Euclid Book 13 Proposition 12
Statement
If an equilateral triangle is inscribed in a circle, then the square of the side of the triangle is triple the square of the radius of the circle.
Computational Explanation
GeometricScene{{A.,B.,C.,O.},{}},{CircleThrough[{A.,B.,C.},O.],GeometricAssertion[{Triangle[{A.,B.,C.}]},"Equilateral"],Line[{A.,O.}]},3
2
EuclideanDistance[A.,B.]
2
EuclideanDistance[A.,O.]
2 EuclideanDistance[A.,B.] 2 EuclideanDistance[A.,O.] |
 |
Explanations
Let be a circle, and let the equilateral triangle be inscribed in it; I say that the square on one side of the triangle is triple of the square on the radius of the circle.
For let the centre of the circle be taken, let be joined and carried through to , and let be joined.
Then, since the triangle is equilateral, therefore the circumference is a third part of the circumference of the circle .
Therefore the circumference is a sixth part of the circumference of the circle; therefore the straight line belongs to a hexagon; therefore it is equal to the radius .
And, since is double of , the square on is quadruple of the square on , that is, of the square on .
But the square on is equal to the squares on , ;therefore the squares on , are quadruple of the square on .
Therefore, separando, the square on is triple of the square on .
But is equal to ; therefore the square on is triple of the square on .
Therefore the square on the side of the triangle is triple of the square on the radius.
ABC
ABC
ABC
For let the centre
D
ABC
AD
E
BE
Then, since the triangle
ABC
BEC
ABC
Therefore the circumference
BE
BE
DE
And, since
AE
DE
AE
ED
BE
But the square on
AE
AB
BE
AB
BE
BE
Therefore, separando, the square on
AB
BE
But
BE
DE
AB
DE
Therefore the square on the side of the triangle is triple of the square on the radius.