Posts in category 'regular-solids'

G.579 segment cut extreme

If a segment be cut in extreme and mean ratio, the square on the greater segment added to the half of the whole is five times the square on the half.

G.580 square segment five

If the square on a segment 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 segment.

G.581 segment cut extreme

If a segment be cut in extreme and mean ratio, the square on the lesser segment added to the half of the greater segment is five times the square on the half of the greater segment.

G.582 segment cut extreme

If a segment 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.

G.583 segment cut extreme

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

G.584 rational segment cut

If a rational segment be cut in extreme and mean ratio, each of the segments is the irrational segment called apotome.

G.585 angles equilateral pentagon

If three angles of an equilateral pentagon, taken either in order or not in order, be equal, the pentagon is equiangular.

G.586 equilateral equiangular pentagon

If in an equilateral and equiangular pentagon segments 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.

G.587 side hexagon decagon

If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole segment has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.

G.588 equilateral pentagon inscribed

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.

G.589 circle diameter rational

If in a circle which has its diameter rational an equilateral pentagon be inscribed, the side of the pentagon is the irrational segment called minor.

G.590 equilateral triangle inscribed

If an :term:`equilateral triangle <G.13>` be :term:`inscribed in a circle <G.148>`, the :term:`square <G.63>` on the side of the :term:`triangle <G.12>` is triple of the square on the radius of the :term:`circle <G.10>`.

G.592 construct pyramid comprehend

To construct a :term:`pyramid <G.553>`, to comprehend it in a given :term:`sphere <G.575>`, and to prove that the :term:`square <G.63>` on the :term:`diameter of the sphere <G.576>` is one and a half times the square on the side of the pyramid.

G.594 construct an octahedron and comprehend it in a

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.

G.595 construct a cube and comprehend it in a

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.

G.597 construct an icosahedron and comprehend it in a

Construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of the icosahedron is the irrational segment called minor.

G.600 construct a dodecahedron and comprehend it in a

Construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures, and to prove that the side of the dodecahedron is the irrational segment called apotome.

G.601 set out sides

To set out the sides of the five figures and to compare them with one another.