— GEOMETOR • elements

G.17 construct an equilateral triangle

Construct an equilateral triangle on a given segment.

G.20 transfer a segment to a point

Construct at a given point a segment equal to a given segment.

G.21 cut off an equal segment

Cut off from the greater of two unequal segments a segment equal to the less.

G.24 congruent triangles by SAS

Triangles with two sides and the included angle equal are congruent (SAS).

G.25 base angles of an isosceles triangle

In an isosceles triangle, the angles at the base are equal, and if the equal sides are extended, the angles under the base are equal.

G.27 triangle with equal base angles is isosceles

If two angles of a triangle are equal, the sides subtending the equal angles are also equal.

G.28 uniqueness of triangle sides on a base

On the same segment and on the same side, there cannot be constructed two distinct triangles with both pairs of conterminous sides equal.

G.29 congruent triangles by SSS

Triangles with all three corresponding sides equal have their corresponding angles equal (SSS).

G.34 construct a perpendicular at a point on a line

Construct a segment at right angles to a given line from a given point on it.

G.35 construct a perpendicular from a point to a line

Construct a perpendicular to a given line from a given point not on it.

G.37 adjacent angles on a line sum to two right angles

A segment standing on a line makes either two right angles or adjacent angles summing to two right angles.

G.39 supplementary adjacent angles form a line

If two segments at a point on a line on opposite sides make adjacent angles summing to two right angles, the two segments are collinear.

G.40 vertical angles are equal

Intersecting lines make their vertical angles equal.

G.41 exterior angle of a triangle is greater than interior

In any triangle, if one side is extended, the exterior angle is greater than either opposite interior angle.

G.42 two angles of a triangle are less than two right angles

In any triangle, the sum of any two angles is less than two right angles.

G.43 greater side subtends greater angle

In any triangle, the greater side subtends the greater angle.

G.44 greater angle is subtended by greater side

In any triangle, the greater angle is subtended by the greater side.

G.45 triangle inequality

In any triangle, the sum of any two sides is greater than the remaining side.

G.46 interior segments of a triangle

Segments constructed from the endpoints of one side of a triangle to an interior point have a lesser sum but contain a greater angle than the other two sides.

G.47 construct a triangle from three segments

Construct a triangle from three segments equal to three given segments satisfying the triangle inequality.

G.48 construct an angle equal to a given angle

Construct on a given line at a given point an angle equal to a given angle.

G.49 hinge theorem

If two triangles have two sides equal to two sides respectively, the one with the greater included angle has the greater base.

G.50 converse hinge theorem

If two triangles have two sides equal to two sides respectively, the one with the greater base has the greater included angle.

G.51 congruent triangles by ASA and AAS

Triangles with two angles and one corresponding side equal are congruent (ASA / AAS).

G.53 equal alternate interior angles imply parallel lines

If a transversal cutting two lines makes the alternate interior angles equal, the lines are parallel.

G.54 equal corresponding angles imply parallel lines

If a transversal cutting two lines makes the exterior angle equal to the opposite interior angle, or interior angles on the same side supplementary, the lines are parallel.

G.56 transversal angles of parallel lines

A transversal cutting parallel lines makes alternate interior angles equal, exterior and opposite interior angles equal, and same-side interior angles supplementary.

G.57 lines parallel to the same line are parallel

Lines parallel to the same line are parallel to one another.

G.58 construct a line parallel to a given line

Construct through a given point a line parallel to a given line.

G.59 exterior angle and angle sum of a triangle

In any triangle, an exterior angle equals the sum of the two opposite interior angles, and the three interior angles sum to two right angles.

G.60 segments joining equal and parallel segments

Segments joining the endpoints of equal and parallel segments in the same directions are themselves equal and parallel.

G.62 opposite sides, angles, and diagonal of a parallelogram

In a parallelogram, opposite sides and opposite angles are equal, and the diagonal bisects the area.

G.64 parallelograms on the same base in the same parallels

Parallelograms on the same base and in the same parallels are equal in area.

G.65 parallelograms on equal bases in the same parallels

Parallelograms on equal bases and in the same parallels are equal in area.

G.66 triangles on the same base in the same parallels

Triangles on the same base and in the same parallels are equal in area.

G.67 triangles on equal bases in the same parallels

Triangles on equal bases and in the same parallels are equal in area.

G.68 equal triangles on the same base are in the same parallels

Equal triangles on the same base and on the same side lie in the same parallels.

G.69 equal triangles on equal bases are in the same parallels

Equal triangles on equal bases and on the same side lie in the same parallels.

G.70 parallelogram is double a triangle on the same base

If a parallelogram shares the same base with a triangle and lies in the same parallels, the parallelogram is double the triangle.

G.71 construct a parallelogram equal to a triangle

Construct, in a given angle, a parallelogram equal in area to a given triangle.

G.72 complements of parallelograms about the diagonal are equal

In any parallelogram, the complements of the parallelograms about the diagonal are equal in area.

G.73 apply a parallelogram equal to a triangle on a segment

Apply to a given segment, in a given angle, a parallelogram equal in area to a given triangle.

G.74 construct a parallelogram equal to a polygon

Construct, in a given angle, a parallelogram equal in area to a given polygon.

G.75 construct a square on a segment

Construct a square on a given segment.

G.79 Pythagorean theorem

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the two legs (Pythagorean Theorem).

G.80 converse of the Pythagorean theorem

If the square on one side of a triangle equals the sum of the squares on the other two sides, the angle contained by those two sides is right.

G.82 distributive law of rectangles

If one of two segments is cut into any number of parts, the rectangle contained by the two segments equals the sum of the rectangles contained by the uncut segment and each of the parts.

G.83 square equals sum of rectangles on parts

If a segment is cut at random, the sum of the rectangles contained by the whole and each of the parts equals the square on the whole.

G.84 rectangle on whole and part

If a segment is cut at random, the rectangle contained by the whole and one of the parts equals the rectangle contained by the parts together with the square on that part.

G.85 square of a sum

If a segment is cut at random, the square on the whole equals the squares on the parts together with twice the rectangle contained by the parts.

G.87 rectangle on unequal parts plus square on difference

If a segment is cut into equal and unequal parts, the rectangle contained by the unequal parts together with the square on the segment between the points of section equals the square on the half.

G.88 rectangle on bisected and extended segment

If a segment is bisected and a segment is added to it in a line, the rectangle contained by the whole with the added segment and the added segment together with the square on the half equals the square on the segment made up of the half and the added segment.

G.89 square of a difference

If a segment is cut at random, the sum of the squares on the whole and on one of the parts equals twice the rectangle contained by the whole and that part together with the square on the remaining part.

G.90 four rectangles plus square on difference

If a segment is cut at random, four times the rectangle contained by the whole and one of the parts together with the square on the remaining part equals the square on the whole and the first part as one segment.

G.91 sum of squares on unequal parts

If a segment is cut into equal and unequal parts, the sum of the squares on the unequal parts is double the sum of the square on the half and the square on the segment between the points of section.

G.92 sum of squares on bisected and extended segment

If a segment is bisected and a segment is added to it in a line, the sum of the squares on the whole with the added segment and on the added segment is double the sum of the square on the half and the square on the half with the added segment.

G.93 cut a segment in extreme and mean ratio

Cut a given segment so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.

G.94 law of cosines for obtuse triangles

In obtuse-angled triangles, the square on the side subtending the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.95 law of cosines for acute triangles

In acute-angled triangles, the square on the side subtending an acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.97 construct a square equal to a polygon

Construct a square equal in area to a given rectilineal figure.

G.98 find the center of a circle

Find the center of a given circle.

G.99 chord of a circle lies within the circle

The segment joining any two points on the arc of a circle falls within the circle.

G.100 diameter bisects chord iff perpendicular

A segment through the center of a circle bisects a chord not through the center if and only if it cuts the chord at right angles.

G.101 intersecting chords not through the center do not bisect

In a circle, two intersecting chords that do not both pass through the center do not bisect one another.

G.102 intersecting circles have distinct centers

Two intersecting circles do not have the same center.

G.104 touching circles have distinct centers

Two circles that touch one another do not have the same center.

G.105 distances to a circle from an eccentric diameter point

From an eccentric point on the diameter of a circle, the greatest segment to the arc passes through the center, the least is the remainder of the diameter, and segments nearer the center are greater than those more remote, with at most two equal segments on opposite sides.

G.106 distances to a circle from an exterior point

From a point outside a circle, of segments falling on the concave arc the greatest passes through the center and nearer ones are greater, while on the convex arc the least lies between the point and the diameter and nearer ones are less, with at most two equal segments.

G.107 point with three equal segments to a circle is the center

A point within a circle from which more than two equal segments fall on the circle is the center of the circle.

G.108 two circles intersect in at most two points

A circle does not intersect another circle at more than two points.

G.109 line of centers of internally touching circles

If two circles touch one another internally, the line joining their centers passes through the point of contact.

G.110 line of centers of externally touching circles

If two circles touch one another externally, the segment joining their centers passes through the point of contact.

G.112 two circles touch at most at one point

A circle does not touch another circle at more than one point, whether internally or externally.

G.114 equal chords are equidistant from the center

In a circle, chords are equal if and only if they are equally distant from the center.

G.116 diameter is the greatest chord

Of chords in a circle, the diameter is the greatest, and a chord nearer the center is always greater than one more remote.

G.118 perpendicular at extremity of a diameter is tangent

The line drawn at right angles to the diameter of a circle from its extremity falls outside the circle, no other line can be interposed between it and the arc, and the angle of the semicircle is greater than any acute rectilineal angle.

G.119 draw a tangent to a circle from a point

From a given point, construct a line tangent to a given circle.

G.120 radius to point of contact is perpendicular to tangent

If a line touches a circle, the segment joined from the center to the point of contact is perpendicular to the tangent.

G.121 perpendicular to tangent at contact passes through center

If a line touches a circle, the line drawn at right angles to the tangent from the point of contact passes through the center of the circle.

G.123 inscribed angle theorem

In a circle, the angle at the center is double the angle at the circumference when the angles have the same arc as base.

G.124 angles in the same circle segment are equal

In a circle, angles in the same segment are equal to one another.

G.125 opposite angles of a cyclic quadrilateral are supplementary

The opposite angles of any quadrilateral inscribed in a circle sum to two right angles.

G.128 no two similar unequal circle segments on the same chord

On the same segment and on the same side of it, there cannot be constructed two similar and unequal segments of circles.

G.129 similar circle segments on equal chords are equal

Similar segments of circles on equal segments are equal to one another.

G.130 complete a circle from a given segment

Given a segment of a circle, complete the full circle of which it is a segment.

G.133 equal angles stand on equal arcs

In equal circles, equal angles stand on equal arcs, whether they stand at the centers or at the circumferences.

G.134 angles standing on equal arcs are equal

In equal circles, angles standing on equal arcs are equal to one another, whether they stand at the centers or at the circumferences.

G.135 equal chords cut off equal arcs

In equal circles, equal chords cut off equal arcs, the greater equal to the greater and the less to the less.

G.136 equal arcs are subtended by equal chords

In equal circles, equal arcs are subtended by equal chords.

G.137 bisect an arc

Bisect a given arc of a circle.

G.139 Thales’ theorem and angles in circle segments

In a circle, the angle in a semicircle is right, the angle in a greater segment is acute, and the angle in a lesser segment is obtuse.

G.140 alternate segment theorem

If a line touches a circle and a chord is drawn from the point of contact, the angles which the chord makes with the tangent equal the angles in the alternate segments of the circle.

G.141 construct a circle segment on a chord admitting a given angle

On a given segment, construct a segment of a circle admitting an angle equal to a given rectilineal angle.

G.142 cut off a circle segment admitting a given angle

From a given circle, cut off a segment admitting an angle equal to a given rectilineal angle.

G.143 intersecting chords theorem

If two chords in a circle intersect one another, the rectangle contained by the segments of one equals the rectangle contained by the segments of the other.

G.144 tangent-secant theorem

If from a point outside a circle a secant and a tangent are drawn, the rectangle contained by the whole secant and its external segment equals the square on the tangent.

G.145 converse of the tangent-secant theorem

If from a point outside a circle a secant and a segment to the circle are drawn such that the rectangle on the whole secant and its external part equals the square on the segment, the segment is tangent to the circle.

G.147 fit a chord in a circle equal to a segment

Fit into a given circle a chord equal to a given segment not greater than the diameter of the circle.

G.149 inscribe a triangle in a circle equiangular with a triangle

Inscribe in a given circle a triangle equiangular with a given triangle.

G.151 circumscribe a triangle about a circle equiangular with a triangle

Circumscribe about a given circle a triangle equiangular with a given triangle.

G.154 inscribe a circle in a triangle

Inscribe a circle in a given triangle.

G.157 circumscribe a circle about a triangle

Circumscribe a circle about a given triangle.

G.158 inscribe a square in a circle

Inscribe a square in a given circle.

G.159 circumscribe a square about a circle

Circumscribe a square about a given circle.

G.160 inscribe a circle in a square

Inscribe a circle in a given square.

G.161 circumscribe a circle about a square

Circumscribe a circle about a given square.

G.162 construct a golden isosceles triangle

Construct an isosceles triangle having each of the angles at the base double the remaining angle.

G.163 inscribe a regular pentagon in a circle

Inscribe an equilateral and equiangular pentagon in a given circle.

G.164 circumscribe a regular pentagon about a circle

Circumscribe an equilateral and equiangular pentagon about a given circle.

G.165 inscribe a circle in a regular pentagon

Inscribe a circle in a given equilateral and equiangular pentagon.

G.166 circumscribe a circle about a regular pentagon

Circumscribe a circle about a given equilateral and equiangular pentagon.

G.167 inscribe a regular hexagon in a circle

Inscribe an equilateral and equiangular hexagon in a given circle.

G.168 inscribe a regular pentadecagon in a circle

Inscribe an equilateral and equiangular fifteen-angled figure in a given circle.

G.171 there number magnitudes

If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.

G.172 magnitude multiple fifth

If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth is the same multiple of the second that the sum of the third and sixth is of the fourth.

G.175 magnitude multiple equimultiples

If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also :term:`ex aequali <G.174>` the magnitudes taken is equimultiples respectively, the one of the second and the other of the fourth.

G.178 magnitude multiple magnitude

If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder is the same multiple of the remainder that the whole is of the whole.

G.179 magnitudes equimultiples magnitudes

If two magnitudes be equimultiples of two magnitudes, and any magnitudes subtracted from them be equimultiples of the same, the remainders also are either equal to the same or equimultiples of them.

G.186 unequal magnitudes greater

Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.

G.193 parts ratio multiples

Parts have the same ratio as the same multiples of them taken in corresponding order.

G.198 magnitudes proportional componendo

If magnitudes be proportional componendo, they is proportional separando.

G.199 magnitudes proportional separando

If magnitudes be proportional separando, they is proportional componendo.

G.201 whole whole part

If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder is to the remainder as whole to whole.

G.207 magnitude ratio fifth

If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth.

G.208 magnitudes proportional greatest

If four magnitudes be proportional, the greatest and the least are greater than the remaining two.

G.210 triangles parallelograms height

Triangles and parallelograms which are under the same height are to one another as their bases.

G.211 segment drawn parallel

If a segment be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut proportionally, the line joining the points of section is parallel to the remaining side of the triangle.

G.212 angle triangle bisected

If an angle of a triangle be bisected and the segment cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if the segments of the base have the same ratio as the remaining sides of the triangle, the segment joined from the vertex to the point of section will bisect the angle of the triangle.

G.213 equiangular triangles sides

In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.

G.214 triangles sides proportional

If two triangles have their sides proportional, the triangles is equiangular and will have those angles equal which the corresponding sides subtend.

G.215 triangles angle equal

If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles is equiangular and will have those angles equal which the corresponding sides subtend.

G.216 triangles angle equal

If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, the triangles is equiangular and will have those angles equal, the sides about which are proportional.

G.218 right angled triangle

If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another.

G.219 cut nth part

From a given segment to cut off a prescribed part.

G.220 cut similarly

Cut a given uncut segment similarly to a given cut segment.

G.221 third proportional

To two given segments to find a third proportional.

G.222 fourth proportional

To three given segments to find a fourth proportional.

G.223 mean proportional

To two given segments to find a mean proportional.

G.225 equal equiangular parallelograms

In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal.

G.226 equal triangles angle

In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal.

G.227 segments proportional rectangle

If four segments be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, the four segments is proportional.

G.229 describe on a given segment a rectilinear figure

Describe on a given segment a rectilinear figure similar and similarly situated to a given rectilinear figure.

G.231 similar triangles duplicate

Similar triangles are to one another in the duplicate ratio of the corresponding sides.

G.232 similar polygons divided

Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side.

G.234 segments proportional polygons

If four segments be proportional, the polygons similar and similarly described upon them is proportional; and, if the polygons similar and similarly described upon them be proportional, the segments will themselves also be proportional.

G.235 equiangular parallelograms ratio

Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.

G.236 parallelogram parallelograms diameter

In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.

G.237 construct one and the same figure similar to

Construct one and the same figure similar to a given polygon and equal to another given polygon.

G.238 parallelogram there taken

If from a parallelogram there be taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, it is about the same diameter with the whole.

G.239 parallelograms applied segment

Of all the parallelograms applied to the same segment and deficient by parallelogrammic figures similar and similarly situated to that described on the half of the segment, that parallelogram is greatest which is applied to the half of the segment and is similar to the defect.

G.240 segment apply parallelogram

To a given segment to apply a parallelogram equal to a given polygon and deficient by a parallelogrammic figure similar to a given one: thus the given polygon must not be greater than the parallelogram described on the half of the segment and similar to the defect.

G.241 segment apply parallelogram

To a given segment to apply a parallelogram equal to a given polygon and exceeding by a parallelogrammic figure similar to a given one.

G.243 extreme and mean ratio

Cut a given segment in extreme and mean ratio.

G.244 right angled triangles

In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle.

G.245 triangles sides proportional

If two triangles having two sides proportional to two sides be placed together at one angle so that their corresponding sides are also parallel, the remaining sides of the triangles is in a segment.

G.246 equal circles angles

In equal circles angles have the same ratio as the arcs on which they stand, whether they stand at the centres or at the arcs.

G.250 unequal numbers set

Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers is prime to one another.

G.251 numbers prime find

Given two numbers not prime to one another, to find their greatest common measure.

G.255 number part parts

Any number is either a part or parts of any number, the less of the greater.

G.257 number part number

If a number be a part of a number, and another be the same part of another, the sum is the same part of the sum that the one is of the one.

G.258 number parts number

If a number be parts of a number, and another be the same parts of another, the sum is the same parts of the sum that the one is of the one.

G.259 number part number

If a number be that part of a number, which a number subtracted is of a number subtracted, the remainder is the same part of the remainder that the whole is of the whole.

G.260 number parts number

If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder is the same parts of the remainder that the whole is of the whole.

G.261 number part number

If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.

G.262 number parts number

If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part will the second also be of the fourth.

G.264 whole whole number

If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder is to the remainder as whole to whole.

G.268 unit measure number

If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth.

G.274 least numbers ratio

The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.

G.282 numbers prime sum

If two numbers be prime to one another, the sum is prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers is prime to one another.

G.291 numbers measure number

If two numbers measure any number, the least number measured by them will also measure the same.

G.345 numbers continued proportion

If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together is prime to the remaining number.

G.350 prime numbers more

Prime numbers are more than any assigned multitude of prime numbers.

G.351 many even numbers

If as many even numbers as we please be added together, the whole is even.

G.353 many odd numbers

If as many odd numbers as we please be added together, and their multitude be even, the whole is even.

G.354 many odd numbers

If as many odd numbers as we please be added together, and their multitude be odd, the whole is odd.

G.355 even number even

If from an even number an even number be subtracted, the remainder is even.

G.356 even number odd

If from an even number an odd number be subtracted, the remainder is odd.

G.357 odd number odd

If from an odd number an odd number be subtracted, the remainder is even.

G.358 odd number even

If from an odd number an even number be subtracted, the remainder is odd.

G.368 many numbers please

If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the excess of the last be to all those before it.

G.370 many numbers please

If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product is perfect.

G.371 unequal magnitudes set

Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there is left some magnitude which is less than the lesser magnitude set out.

G.373 less unequal magnitudes

If, when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes is incommensurable.

G.374 commensurable magnitudes find

Given two commensurable magnitudes, to find their greatest common measure.

G.385 magnitudes commensurable incommensurable

If two magnitudes be commensurable, and the one of them be incommensurable with any magnitude, the remaining one is incommensurable with the same.

G.387 commensurable magnitudes added

If two commensurable magnitudes be added together, the whole is commensurable with each of them; and, if the whole be commensurable with one of them, the original magnitudes is commensurable.

G.388 incommensurable magnitudes added

If two incommensurable magnitudes be added together, the whole is incommensurable with each of them; and, if the whole be incommensurable with one of them, the original magnitudes is incommensurable.

G.389 there unequal segments

If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are commensurable in length, then the square on the greater is greater than the square on the less by the square on a segment commensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment commensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it will divide it into parts which are commensurable in length.

G.390 there unequal segments

If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are incommensurable, the square on the greater is greater than the square on the less by the square on a segment incommensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment incommensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it divides it into parts which are incommensurable.

G.393 rectangle contained rational

The rectangle contained by rational segments commensurable in length is rational.

G.394 rational area applied

If a rational area be applied to a rational segment, it produces as breadth a segment rational and commensurable in length with the segment to which it is applied.

G.395 rectangle contained rational

The rectangle contained by rational segments commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial.

G.396 square medial segment

The square on a medial segment, if applied to a rational segment, produces as breadth a segment rational and incommensurable in length with that to which it is applied.

G.397 segment commensurable medial

A segment commensurable with a medial segment is medial.

G.398 rectangle contained medial

The rectangle contained by medial segments commensurable in length is medial.

G.399 rectangle contained medial

The rectangle contained by medial segments commensurable in square only is either rational or medial.

G.400 medial area exceed

A medial area does not exceed a medial area by a rational area.

G.403 find two rational segments commensurable in square only

Find two rational segments commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a segment commensurable in length with the greater.

G.404 find two rational segments commensurable in square only

Find two rational segments commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a segment incommensurable in length with the greater.

G.407 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial.

G.408 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.

G.409 find two segments incommensurable in square which make

Find two segments incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them.

G.410 rational segments commensurable

If two rational segments commensurable in square only be added together, the whole is irrational; and let it be called binomial.

G.411 medial segments commensurable

If two medial segments commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial segment.

G.412 medial segments commensurable

If two medial segments commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a second bimedial segment.

G.413 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole segment is irrational: and let it be called major.

G.414 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational, be added together, the whole segment is irrational; and let it be called the side of a rational plus a medial area.

G.415 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them medial, and the rectangle contained by them medial and also incommensurable with the sum of the squares on them, be added together, the whole segment is irrational; and let it be called the side of the sum of two medial areas.

G.416 binomial segment divided

A binomial segment is divided into its terms at one point only.

G.417 bimedial segment divided

A first bimedial segment is divided at one point only.

G.418 bimedial segment divided

A second bimedial segment is divided at one point only.

G.419 major segment divided

A major segment is divided at one and the same point only.

G.420 side rational plus

The side of a rational plus a medial area is divided at one point only.

G.421 side sum medial

The side of the sum of two medial areas is divided at one point only.

G.423 find the first binomial segment

Find the first binomial segment.

G.425 find the second binomial segment

Find the second binomial segment.

G.427 find the third binomial segment

Find the third binomial segment.

G.429 find the fourth binomial segment

Find the fourth binomial segment.

G.431 find the fifth binomial segment

Find the fifth binomial segment.

G.433 find the sixth binomial segment

Find the sixth binomial segment.

G.434 area contained rational

If an area be contained by a rational segment and the first binomial, the “side” of the area is the irrational segment which is called binomial.

G.435 area contained rational

If an area be contained by a rational segment and the second binomial, the “side” of the area is the irrational segment which is called a first bimedial.

G.436 area contained rational

If an area be contained by a rational segment and the third binomial, the “side” of the area is the irrational segment called a second bimedial.

G.437 area contained rational

If an area be contained by a rational segment and the fourth binomial, the “side” of the area is the irrational segment called major.

G.438 area contained rational

If an area be contained by a rational segment and the fifth binomial, the “side” of the area is the irrational segment called the side of a rational plus a medial area.

G.439 area contained rational

If an area be contained by a rational segment and the sixth binomial, the “side” of the area is the irrational segment called the side of the sum of two medial areas.

G.440 square binomial segment

The square on the binomial segment applied to a rational segment produces as breadth the first binomial.

G.441 square bimedial segment

The square on the first bimedial segment applied to a rational segment produces as breadth the second binomial.

G.442 square bimedial segment

The square on the second bimedial segment applied to a rational segment produces as breadth the third binomial.

G.443 square major segment

The square on the major segment applied to a rational segment produces as breadth the fourth binomial.

G.444 square side rational

The square on the side of a rational plus a medial area applied to a rational segment produces as breadth the fifth binomial.

G.445 square side sum

The square on the side of the sum of two medial areas applied to a rational segment produces as breadth the sixth binomial.

G.446 segment commensurable length

A segment commensurable in length with a binomial segment is itself also binomial and the same in order.

G.447 segment commensurable length

A segment commensurable in length with a bimedial segment is itself also bimedial and the same in order.

G.448 segment commensurable major

A segment commensurable with a major segment is itself also major.

G.449 segment commensurable side

A segment commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.

G.450 segment commensurable side

A segment commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas.

G.451 rational medial area

If a rational and a medial area be added together, four irrational segments arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area.

G.452 medial areas incommensurable

If two medial areas incommensurable with one another be added together, the remaining two irrational segments arise, namely either a second bimedial or a side of the sum of two medial areas.

G.453 rational segment there

If from a rational segment there be subtracted a rational segment commensurable with the whole in square only, the remainder is irrational; and let it be called an apotome.

G.455 medial segment there

If from a medial segment there be subtracted a medial segment which is commensurable with the whole in square only, and which contains with the whole a rational rectangle, the remainder is irrational. And let it be called a first apotome of a medial segment.

G.457 medial segment there

If from a medial segment there be subtracted a medial segment which is commensurable with the whole in square only, and which contains with the whole a medial rectangle, the remainder is irrational; and let it be called a second apotome of a medial segment.

G.458 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole and which with the whole makes the squares on them added together rational, but the rectangle contained by them medial, the remainder is irrational; and let it be called minor.

G.459 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, but twice the rectangle contained by them rational, the remainder is irrational: and let it be called that which produces with a rational area a medial whole.

G.460 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them, the remainder is irrational; and let it be called that which produces with a medial area a medial whole.

G.461 apotome only rational

To an apotome only one rational segment can be annexed which is commensurable with the whole in square only.

G.462 apotome medial segment

To a first apotome of a medial segment only one medial segment can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle.

G.463 apotome medial segment

To a second apotome of a medial segment only one medial segment can be annexed which is commensurable with the whole in square only and which contains with the whole a medial rectangle.

G.464 minor segment only

To a minor segment only one segment can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of the squares on them rational but twice the rectangle contained by them medial.

G.465 segment produces rational

To a segment which produces with a rational area a medial whole only one segment can be annexed which is incommensurable in square with the whole segment and which with the whole segment makes the sum of the squares on them medial, but twice the rectangle contained by them rational.

G.466 segment produces medial

To a segment which produces with a medial area a medial whole only one segment can be annexed which is incommensurable in square with the whole segment and which with the whole segment makes the sum of the squares on them medial and twice the rectangle contained by them both medial and also incommensurable with the sum of the squares on them.

G.467 find the first apotome

Find the first apotome.

G.468 find the second apotome

Find the second apotome.

G.470 find the third apotome

Find the third apotome.

G.472 find the fourth apotome

Find the fourth apotome.

G.474 find the fifth apotome

Find the fifth apotome.

G.476 find the sixth apotome

Find the sixth apotome.

G.477 area contained rational

If an area be contained by a rational segment and a first apotome, the “side” of the area is an apotome.

G.478 area contained rational

If an area be contained by a rational segment and a second apotome, the “side” of the area is a first apotome of a medial segment.

G.479 area contained rational

If an area be contained by a rational segment and a third apotome, the “side” of the area is a second apotome of a medial segment.

G.480 area contained rational

If an area be contained by a rational segment and a fourth apotome, the “side” of the area is minor.

G.481 area contained rational

If an area be contained by a rational segment and a fifth apotome, the “side” of the area is a segment which produces with a rational area a medial whole.

G.482 area contained rational

If an area be contained by a rational segment and a sixth apotome, the “side” of the area is a segment which produces with a medial area a medial whole.

G.483 square apotome applied

The square on an apotome applied to a rational segment produces as breadth a first apotome.

G.484 square apotome medial

The square on a first apotome of a medial segment applied to a rational segment produces as breadth a second apotome.

G.485 square apotome medial

The square on a second apotome of a medial segment applied to a rational segment produces as breadth a third apotome.

G.486 square minor segment

The square on a minor segment applied to a rational segment produces as breadth a fourth apotome.

G.487 square segment produces

The square on the segment which produces with a rational area a medial whole, if applied to a rational segment, produces as breadth a fifth apotome.

G.488 square segment produces

The square on the segment which produces with a medial area a medial whole, if applied to a rational segment, produces as breadth a sixth apotome.

G.489 segment commensurable length

A segment commensurable in length with an apotome is an apotome and the same in order.

G.490 segment commensurable apotome

A segment commensurable with an apotome of a medial segment is an apotome of a medial segment and the same in order.

G.491 segment commensurable minor

A segment commensurable with a minor segment is minor.

G.492 segment commensurable produces

A segment commensurable with that which produces with a rational area a medial whole is a segment which produces with a rational area a medial whole.

G.493 segment commensurable produces

A segment commensurable with that which produces with a medial area a medial whole is itself also a segment which produces with a medial area a medial whole.

G.494 rational area medial

If from a rational area a medial area be subtracted, the “side” of the remaining area becomes one of two irrational segments, either an apotome or a minor segment.

G.495 medial area rational

If from a medial area a rational area be subtracted, there arise two other irrational segments, either a first apotome of a medial segment or a segment which produces with a rational area a medial whole.

G.496 medial area there

If from a medial area there be subtracted a medial area incommensurable with the whole, the two remaining irrational segments arise, either a second apotome of a medial segment or a segment which produces with a medial area a medial whole.

G.497 apotome binomial segment

The apotome is not the same with the binomial segment.

G.498 square rational segment

The square on a rational segment applied to the binomial segment produces as breadth an apotome the terms of which are commensurable with the terms of the binomial and moreover in the same ratio; and further the apotome so arising will have the same order as the binomial segment.

G.499 square rational segment

The square on a rational segment, if applied to an apotome, produces as breadth the binomial segment the terms of which are commensurable with the terms of the apotome and in the same ratio; and further the binomial so arising has the same order as the apotome.

G.500 area contained apotome

If an area be contained by an apotome and the binomial segment the terms of which are commensurable with the terms of the apotome and in the same ratio, the “side” of the area is rational.

G.502 part segment plane

A part of a segment cannot be in the plane of reference and a part in a plane more elevated.

G.503 segments cut plane

If two segments cut one another, they are in one plane, and every triangle is in one plane.

G.504 planes cut common

If two planes cut one another, their common section is a segment.

G.506 segment set right

If a segment be set up at right angles to two segments which cut one another, at their common point of section, it is at right angles to the plane through them.

G.507 segment set right

If a segment be set up at right angles to three segments which meet one another, at their common point of section, the three segments are in one plane.

G.508 segments right angles

If two segments be at right angles to the same plane, the segments is parallel.

G.509 segments parallel points

If two segments be parallel and points be taken at random on each of them, the segment joining the points is in the same plane with the parallel lines.

G.510 segments parallel right

If two segments be parallel, and one of them be at right angles to any plane, the remaining one is at right angles to the same plane.

G.511 straight lines parallel

Straight lines which are parallel to the same segment and are not in the same plane with it are also parallel to one another.

G.512 segments meeting parallel

If two segments meeting one another be parallel to two segments meeting one another not in the same plane, they will contain equal angles.

G.513 elevated point draw

From a given elevated point to draw a segment perpendicular to a given plane.

G.514 set segment right

To set up a segment at right angles to a given plane from a given point in it.

G.515 point segments set

From the same point two segments cannot be set up at right angles to the same plane on the same side.

G.517 planes segment right

Planes to which the same segment is at right angles is parallel.

G.518 segments meeting parallel

If two segments meeting one another be parallel to two segments meeting one another, not being in the same plane, the planes through them are parallel.

G.519 parallel planes cut

If two parallel planes be cut by any plane, their common sections are parallel.

G.520 segments cut parallel

If two segments be cut by parallel planes, they is cut in the same ratios.

G.522 segment right angles

If a segment be at right angles to any plane, all the planes through it is at right angles to the same plane.

G.523 planes cut right

If two planes which cut one another be at right angles to any plane, their common section is at right angles to the same plane.

G.525 solid angle contained

If a solid angle be contained by three plane angles, any two, taken together in any manner, are greater than the remaining one.

G.526 solid angle contained

Any solid angle is contained by plane angles less than four right angles.

G.527 there plane angles

If there be three plane angles of which two, taken together in any manner, are greater than the remaining one, and they are contained by equal segments, it is possible to construct a triangle out of the segments joining the extremities of the equal segments.

G.528 construct a solid angle out of three plane

Construct a solid angle out of three plane angles two of which, taken together in any manner, are greater than the remaining one: thus the three angles must be less than four right angles.

G.530 solid contained parallel

If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.

G.531 parallelepipedal solid cut

If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.

G.532 segment point construct

On a given segment, and at a given point on it, to construct a solid angle equal to a given solid angle.

G.536 describe on a given segment a parallelepipedal solid

Describe on a given segment a parallelepipedal solid similar and similarly situated to a given parallelepipedal solid.

G.538 parallelepipedal solid cut

If a parallelepipedal solid be cut by a plane through the diagonals of the opposite planes, the solid is bisected by the plane.

G.539 parallelepipedal solids base

Parallelepipedal solids which are on the same base and of the same height, and in which the extremities of the sides which stand up are on the same segments, are equal.

G.540 parallelepipedal solids base

Parallelepipedal solids which are on the same base and of the same height, and in which the extremities of the sides which stand up are not on the same segments, are equal.

G.541 parallelepipedal solids equal

Parallelepipedal solids which are on equal bases and of the same height are equal.

G.542 parallelepipedal solids height

Parallelepipedal solids which are of the same height are to one another as their bases.

G.543 similar parallelepipedal solids

Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides.

G.544 equal parallelepipedal solids

In equal parallelepipedal solids the bases are reciprocally proportional to the heights; and those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal.

G.545 there equal plane

If there be two equal plane angles, and on their vertices there be set up elevated segments containing equal angles with the original segments respectively, if on the elevated segments points be taken at random and perpendiculars be drawn from them to the planes in which the original angles are, and if from the points so arising in the planes segments be joined to the vertices of the original angles, they will contain, with the elevated segments, equal angles.

G.546 segments proportional parallelepipedal

If three segments be proportional, the parallelepipedal solid formed out of the three is equal to the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid.

G.547 segments proportional parallelepipedal

If four segments be proportional, the parallelepipedal solids on them which are similar and similarly described is proportional; and, if the parallelepipedal solids on them which are similar and similarly described be proportional, the segments will themselves also be proportional.

G.549 sides opposite planes

If the sides of the opposite planes of a cube be bisected, and planes be carried through the points of section, the common section of the planes and the diameter of the cube bisect one another.

G.551 similar polygons inscribed

Similar polygons inscribed in circles are to one another as the squares on the diameters.

G.552 circles squares diameters

Circles are to one another as the squares on the diameters.

G.554 pyramid triangular base

Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.

G.555 there pyramids height

If there be two pyramids of the same height which have triangular bases, and each of them be divided into two pyramids equal to one another and similar to the whole, and into two equal prisms, then, as the base of the one pyramid is to the base of the other pyramid, so will all the prisms in the one pyramid be to all the prisms, being equal in multitude, in the other pyramid.

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