— GEOMETOR • elements

G.127 similar circle segments

Similar segments of circles are those which admit equal:term:angles <G.22>, or in which the angles are equal.

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.153 circle inscribed in a polygon

Similarly a:term:circle <G.10> is said to be:term:**inscribed in a figure** <circle-inscribed-in-figure> when the arc of the circle touches each side of the:term:figure <G.9> in which it is inscribed.

G.155 polygon circumscribed about a polygon

Similarly a:term:figure <G.9> is said to be:term:**circumscribed about a figure** <circumscribed-about-figure> when the respective sides of the circumscribed figure pass through the respective:term:angles <G.22> of that about which it is circumscribed.

G.217 similar rectilineal figures

Similar polygons are such as have their:term:angles <G.22> severally equal and the sides about the equal angles:term:proportional <G.180>.

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.220 cut similarly

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

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.233 figures similar polygon

Figures which are similar to the same polygon are also similar to one another.

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.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.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.318 similar plane and solid numbers

Similar plane and solid numbers are those which have their sides:term:proportional <G.180>.

G.319 similar plane numbers

Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.

G.321 similar solid numbers

Between two similar solid numbers there fall two mean proportional numbers; and the solid number has to the similar solid number the ratio triplicate of that which the corresponding side has to the corresponding side.

G.322 mean proportional number

If one mean proportional number fall between two numbers, the numbers is similar plane numbers.

G.323 mean proportional numbers

If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.

G.328 similar plane numbers

Similar plane numbers have to one another the ratio which a square number has to a square number.

G.329 similar solid numbers

Similar solid numbers have to one another the ratio which a cube number has to a cube number.

G.330 similar plane numbers

If two similar plane numbers by multiplying one another make some number, the product is square.

G.331 numbers multiplying make

If two numbers by multiplying one another make a square number, they are similar plane numbers.

G.534 similar solid figures

Similar solid figures are those:term:contained by similar planes <G.533> equal in multitude.

G.535 equal and similar solid figures

Equal and similar solid figures are those:term:contained by similar planes <G.533> equal in multitude and in magnitude.

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.537 prism

A:term:**prism** <prism> is a:term:solid figure <G.529> contained by:term:planes <G.7> two of which, namely those which are opposite, are equal, similar and:term:parallel <G.52>, while the rest are parallelograms.

G.543 similar parallelepipedal solids

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

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.551 similar polygons inscribed

Similar polygons inscribed in 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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