— GEOMETOR • elements

G.12 rectilineal figures

Rectilineal figures are contained by segments: trilateral by three, quadrilateral by four, and multilateral by more than four.

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.63 square, oblong, rhombus, and rhomboid

Of quadrilateral figures, a square is equilateral and right-angled; an oblong is right-angled but not equilateral; a rhombus is equilateral but not right-angled; a rhomboid has opposite sides and angles equal but is neither equilateral nor right-angled.

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.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.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.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.97 construct a square equal to a polygon

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

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

Circumscribe about a given circle a triangle equiangular with 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.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.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.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.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.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.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.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.503 segments cut plane

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

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.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.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.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.

Posts tagged 'quadrilateral'