G.594 construct an octahedron and comprehend it in a

XIII.14

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.

construct_octahedron_comprehend(/ A B /, C)

given the diameter / A B / of the given sphere be set out, let it be bisected at C

construct the octahedron has been comprehended in the given sphere, and it has been demonstrated at the same time that the square on the diameter of the sphere is double of the square on the side of the octahedron

given the diameter / A B / of the given sphere be set out, let it be bisected at C

construct Let the semicircle ( A D B ) be described on / A B /

construct perpendicular_at(/ A B /, C) → / C D /

construct segment(D, B) → / D B /

construct segment() → < E F G H >, segment(D, B) → / D B /, segment(H, F) → / H F /, segment(E, G) → / E G /

construct set_segment_right: From K let / K L / be set up at right angles to the plane of the square < E F G H >, and let it be carried through to the other side of the plane, as / K M /

construct From / K L /, / K M / let / K L /, / K M / be respectively cut off equal to one of / E K /, / F K /, / G K /, / H K /, and let / L E /, / L F /, / L G /, / L H /, / M E /, / M F /, / M G /, / M H / be joined.

∵ pythagorean_theoremassert / K E / = / K H /, and < E K H ) is right, therefore the square on / H E / is double of the square on / E K /

∵ pythagorean_theoremassert / L K / = / K E /, and < L K E ) is right, therefore the square on / E L / is double of the square on / E K /

assert The square on / H E / was also proved double of the square on / E K /

assert the square on / L E / = the square on / E H /

assert / L E / = / E H /

assert The same reason / L H / = / H E /

assert < L E H > is equilateral

assert Similarly we can prove that each of the remaining triangles of which the sides of the square < E F G H > are the bases, and L, M the vertices, is equilateral

assert an octahedron has been constructed which is contained by eight equilateral triangles

assert It is next required to comprehend it in the given sphere, and to prove that the square on the diameter of the sphere is double of the square on the side of the octahedron

assert / L K /, / K M /, / K E / =, therefore the semicircle described on / L M / will also pass through E

assert For the same reason, if, / L M / remaining fixed, the semicircle be carried round and restored to the same position from which it began to be moved, it will also pass through F, G, H, and the octahedron will have been comprehended in a sphere

assert I say next that it is also comprehended in the given sphere

∵ sas_congruenceassert / L K / = / K M /, while / K E / is common, and they contain right angles, therefore / L E / = / E M /

∵ thales_semicircle_theorem, pythagorean_theoremassert since < L E M ) is right, for it is in a semicircle, therefore the square on / L M / is double of the square on / L E /

assert / A C / = / C B /, / A B / is double of / B C /

assert But, as / A B / is to / B C /, so is the square on / A B / to the square on / B D /

assert the square on / A B / is double of the square on / B D /

assert The square on / L M / was also proved double of the square on / L E /

assert The square on / D B / = the square on / L E /, for / E H / was made = to / D B /

assert the square on / A B / = the square on / L M /

assert / A B / = / L M /

assert / A B / is the diameter of the given sphere

assert / L M / = the diameter of the given sphere

∴ the octahedron has been comprehended in the given sphere, and it has been demonstrated at the same time that the square on the diameter of the sphere is double of the square on the side of the octahedron

Dependency Graph

Required for

G.601