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Answer 1

Here 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.

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   --------------------(i)

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} -----(ii).

Substitute equation (ii) in equation (i), 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.