Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangle drawn on the other two sides of the triangle.

Let a right triangle QPR in which ÐP is right angle and PR = y, PQ = x.

Three equilateral triangles ∆PER, ∆PFR and ∆RQD are drawn on the three sides of ∆PQR.

Again, let area of triangles made on PR, PQ are A1, A2 and A3, respectively.

To prove A₃ = A₁ + A₂

In ∆RPQ,

By Pythagoras theorem,

QR² = PR² + PQ²

⇒ QR² = y² + x²

We know that,

Area of an equilateral triangle =

∴ Area of equilateral ∆PER,

And area of equilateral ∆PFQ,

A₁ + A₂

[from Equation (i) and (ii)]

Hence proved.