Question

Prove that the area of an equilateral triangle described on the diagonal of a square is twice the area of an equilateral triangle described on its side.

Answer 1

Let the side of Square be $a$

Diagonal $a\sqrt{2}$

Area of equilateral triangle on its Diagonal is $\frac{\sqrt{3}}{4}(a\sqrt{2}{)}^2\frac{\sqrt{3}}{2}{a}^2\cdots (1)$

area of equilateral triangle in its side is $\frac{\sqrt{3}}{4}{a}^2\cdots \left(2\right)$

$\left(1\right)=\left(2\right)$

Hence proved