Question

P, Q and R are the mid points of sides BC, CA, and AB of ABC, and AD is. the perpendicular from A to BC.Prove that P, Q, R and D are concyclic.

Answer 1

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Answer 2

As R and Q are mid points of AB and CA, therefore

RQ || BC ( By mid point theorem)

Similarly,  PR || CA and PQ || AB

In quadrilateral BPQR,

BP || RQ and PQ || BR (

Thus, BPQR is a parallelogram.

Similarly, ARPQ is a parallelogram.

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

∠BPR = ∠C (Corresponding angles)

∠DPQ = ∠RPQ + ∠DPR  = ∠A + ∠C .--- eq 1

∠ARO = ∠B (Corresponding angles) --- eq 2

In ΔABD, the mid point of AB is R

Since, OR || BD, thus O is the mid point of AD (Converse of the mid point theorem)

= OA = OD

In ΔAOR and ΔDOR,

OA = OD (Proved)

OR = OR (Common)

∠AOR = ∠DOR (90°) ( Linear Pair)

ΔAOR ≅ ΔDOR (SAS congruence )

= ∠ARO = ∠DRO (CPCT) and ∠DRO = ∠B (Using (2))

In quadrilateral PRQD,

∠DRO + ∠DPQ = ∠B + ( ∠A + ∠C) By 1

= ∠DRO + ∠DPQ = 180°

Hence, quadrilateral PRQD is a cyclic quadrilateral and the points P, Q, R and D are concyclic.