Given:

• Square ABCD

• Equilateral Triangles BEC and AFC

In ∆BEC and ∆AFC,

∠BEC = ∠AFC = 60°

∠BCE = ∠ACF = 60°

(∵ ∆BEC and ∆AFC both are equilateral triangles)

So, by AA rule,

∆BEC ~ ∆AFC

So,

(∵ Ratio of areas of similar triangles equals to the ratio of the square of their corresponding sides)

Now,

Hence, the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonals.

OR

Given:

• ∆ ABC ∆ PQR

• Ar(∆ ABC) = Ar(∆ PQR)

We know,

(∵ Ratio of areas of similar triangles equals to the ratio of the square of their corresponding sides)

⇒ BC = QR and AB = PQ AC = PR

∴ ∆ ABC ≅ ∆ PQR by SSS rule.