In an equilateral triangle ABC; points P, Q and R are taken on the sides AB, BC and CA respectively such that AP = BQ = CR. Prove that triangle PQR is equilateral.

Given,

ABC is an equilateral triangle.

∴ AB = BC = CA ……..(1)

Given,

⇒ AP = BQ = CR ……….(2)

Subtracting equation (2) from (1), we get :

⇒ AB - AP = BC - BQ = CA - CR

⇒ BP = CQ = AR ……….(3)

Since, ABC is an equilateral triangle,

∴ ∠A = ∠B = ∠C ……..(4)

In △ BPQ and △ CQR,

⇒ BP = CQ [From equation (3)]

⇒ ∠B = ∠C [From equation (4)]

⇒ BQ = CR [Given]

∴ △ BPQ ≅ △ CQR (By S.A.S. axiom)

We know that,

Corresponding parts of congruent triangle are equal.

⇒ PQ = QR …….(5)

In △ CQR and △ APR,

⇒ CQ = AR [From equation (3)]

⇒ ∠C = ∠A [From equation (4)]

⇒ CR = AP [Given]

∴ △ CQR ≅ △ APR (By S.A.S. axiom)

∴ QR = PR (By C.P.C.T.C.) ……..(6)

From equation (5) and (6),

PQ = QR = PR.

Hence, proved that PQR is an equilateral triangle.