Question

If P, Q and R are the mid-points of the sides BC, CA and AB of a triangle and AD is the perpendicular from A on BC, prove that P, Q, R and D are concyclic.

Answer 1

Given: In ΔABC, P, Q and R are the mid-points of the sides BC, CA and AB respectively. Also, AD ⊥ BC.

To prove: P, Q, R and D are concyclic.

Construction: Join DR, RQ and QP

Proof: In right-angled ΔADP, R is the mid-point of AB.

∴ RB = RD

⇒ ∠2 = ∠1  ...(i) [Angles opposite to the equal sides are equal]

Since, R and Q are the mid-points of AB and AC, then

RQ || BC  ...[By mid-point theorem]

or RQ || BP

Since, QP || RB, then quadrilateral BPQR is a parallelogram.

⇒ ∠1 = ∠3   ...(ii) [Opposite angles of parallelogram are equal]

From equations (i) and (ii),

∠2 = ∠3

But ∠2 + ∠4 = 180°   ...[Linear pair axiom]

∴ ∠3 + ∠4 = 180°  ...[∴ ∠2 = ∠3]

Hence, quadrilateral PQRD is a cyclic quadrilateral.

So, points P, Q, R and D are non-cyclic.

Hence proved.