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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.
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Solution
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.
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