P,Q,R are mid points of  BC …

Ans.

Given : In ΔABC, P, Q and R are the mid points of sides BC, CA and AB respectively. AD ⊥ BC.
To prove : P, Q, R and D are concyclic.

Proof : In ΔABC, R and Q are mid points of AB and CA respectively.

∴ RQ || BC (Mid point theorem)

Similarly, PQ || AB and PR || CA

In quadrilateral BPQR,

BP || RQ and PQ || BR        [RQ || BC and PQ || AB]

∴ Quadrilateral BPQR is a parallelogram.

Similarly, quadrilateral ARPQ is a parallelogram

∴ ∠A = ∠RPQ (Opposite sides of parallelogram are equal)

PR || AC and PC is the transversal

∴ ∠BPR = ∠C (Corresponding angles)

∠DPQ = ∠DPR + ∠RPQ = ∠A + ∠C ...           (1)

RQ || BC and BR is the transversal,

∴ ∠ARO = ∠B (Corresponding angles) ...          (2)

In ΔABD, R is the mid point of AB and OR || BD.

∴ O is the mid point of AD (Converse of mid point theorem)

⇒ OA = OD

In ΔAOR and ΔDOR

OA = OD (Proved)

∠AOR = ∠DOR           (90°) [∠ROD = ∠ODP (Alternate angles)  & ∠AOR = ∠ROD = 90° (linear pair)]

OR = OR (Common)

∴ ΔAOR  ≅ ΔDOR (SAS congruence criterion)

⇒ ∠ARO = ∠DRO (CPCT)

⇒ ∠DRO = ∠B (Using (2))

In quadrilateral PRQD,

∠DRO + ∠DPQ = ∠B + ( ∠A + ∠C) = ∠A + ∠B + ∠C (Using (1))

⇒ ∠DRO + ∠DPQ = 180° ( ∠A + ∠B + ∠C = 180°)

Hence, quadrilateral PRQD is a cyclic quadrilateral.