G.590 equilateral triangle inscribed — Heath

XIII.12

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>`.

Let ( A B C ) be a circle, and let the equilateral triangle < A B C > be inscribed in it; I say that the square on one side of the triangle < A B C > is triple of the square on the radius of the circle.

For let the centre D of the circle ( A B C ) be taken, let / A D / be joined and carried through to E, and let / B E / be joined.

Then, since the triangle < A B C > is equilateral,

therefore the circumference ( B E C ) is a third part of the circumference of the circle ( A B C ).

Therefore the circumference ( B E ) is a sixth part of the circumference of the circle; therefore the straight line / B E / belongs to a hexagon; therefore it is equal to the radius / D E /. [G.167]

And, since / A E / is double of / D E /,

the square on / A E / is quadruple of the square on / E D /, that is, of the square on / B E /.

But the square on / A E / is equal to the squares on / A B /, / B E /; [G.139, G.79]

therefore the squares on / A B /, / B E / are quadruple of the square on / B E /.

Therefore, separando, the square on / A B / is triple of the square on / B E /.

But / B E / is equal to / D E /;

therefore the square on / A B / is triple of the square on / D E /.

Therefore the square on the side of the triangle is triple of the square on the radius.

Q. E. D.

Dependency Graph

Required for

G.592, G.601