— GEOMETOR • elements

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.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.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.74 construct a parallelogram equal to a polygon

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

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.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.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.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.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.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.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.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.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.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.167 inscribe a regular hexagon in a circle

Inscribe an equilateral and equiangular hexagon in a given circle.

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.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.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.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.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.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.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.503 segments cut plane

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

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.517 planes segment right

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

G.520 segments cut parallel

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

G.526 solid angle contained

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

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.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.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.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.550 there prisms equal

If there be two prisms of equal height, and one have a parallelogram as base and the other a triangle, and if the parallelogram be double of the triangle, the prisms is equal.

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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