Question

prove that area of equilateral triangle to solve on side of square equal to half of the area of equilateral triangle described one of its diagonal

Answer 1

Hope this helps you........

Answer 2

ABCD is a square, AEB is an equilateral triangle described on the side of the square and DBF is an equilateral triangle described on diagonal BD of  the square.
We have to Prove: Ar(ΔDBF) / Ar(ΔAEB) = 2 / 1 
Proof:  If two equilateral triangles are similar then all angles are = 60 degrees.

Therefore, by AAA similarity criterion , △DBF ~

 △AEB

Ar(ΔDBF) / Ar(ΔAEB) = DB2 /AB2 -----------1

We know that the ratio of the areas of two similar triangles is equal to
the square of the ratio of their corresponding sides i .e.
But, we have DB = √2AB     {But diagonal of square is √2 times of its side} -----2
Substitute equation 2 in equation 1, we get
Ar(ΔDBF) / Ar(ΔAEB) = (√2AB )2 / AB2   = 2 AB2 / AB2 = 2

∴ Area of equilateral triangle described on one side os square is equal to half the area of the equilateral triangle described on one of its diagonals.