BOOK XIII. PROPOSITIONS.

L F N M H O A C B D K E G

Proposition XIII.2

If the square on a straight line be five times the square on a segment of it, then, when the double of the said segment is cut in extreme and mean ratio, the greater segment is the remaining part of the original straight line.
A C B H L M F N K D G E

Proposition XIII.4

If a straight line be cut in extreme and mean ratio, the square on the whole and the square on the lesser segment together are triple of the square on the greater segment.
D A C B L H K E

Proposition XIII.5

If a straight line be cut in extreme and mean ratio, and there be added to it a straight line equal to the greater segment, the whole straight line has been cut in extreme and mean ratio, and the original straight line is the greater segment.
D A C B

Proposition XIII.6

If a rational straight line be cut in extreme and mean ratio, each of the segments is the irrational straight line called apotome.
A B F E C D

Proposition XIII.7

If three angles of an equilateral pentagon, taken either in order or not in order, be equal, the pentagon will be equiangular.
A B C D E H

Proposition XIII.8

If in an equilateral and equiangular pentagon straight lines subtend two angles taken in order, they cut one another in extreme and mean ratio, and their greater segments are equal to the side of the pentagon.
A B C D E F

Proposition XIII.9

If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.
A B C D E F G H K L M N

Proposition XIII.10

If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the squares on the side of the hexagon and on that of the decagon inscribed in the same circle.
A B C D E

Proposition XIII.12

If an :term:`equilateral triangle <equilateral-triangle>` be :term:`inscribed in a circle <inscribed-in-circle>`, the :term:`square <square>` on the side of the :term:`triangle <trilateral>` is triple of the square on the radius of the :term:`circle <circle>`.
A B C D E F G H K L

Proposition XIII.13

To construct a :term:`pyramid <pyramid>`, to comprehend it in a given :term:`sphere <sphere>`, and to prove that the :term:`square <square>` on the :term:`diameter of the sphere <diameter-of-sphere>` is one and a half times the square on the side of the pyramid.
A B C D L E H K F G M

Proposition XIII.14

To construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the diameter of the sphere is double of the square on the side of the octahedron.
A B C D E H K N F G L M

Proposition XIII.15

To construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple of the square on the side of the cube.