Prove that area of an equilateral triangle formed an one side of a square is half of the area of the equilateral triangle formed on one diagonal of that square itself.

Let there be a square ABCD with diagonal AC of side ‘a’ .

For equilateral triangle drawn on one side of the square,

In ΔB₁C₁E₁,

Side = a

For the equilateral triangle formed on one diagonal of that square,

In ΔABC,

side = √2a

Thus, the area of an equilateral triangle formed an one side of a square is half of the area of the equilateral triangle formed on one diagonal of that square itself.

Hence, proved.