in the triangle abc ad perpendicular to bc br perpendicular to ac and aq perpendicular to ab then show that angle opq = angle opr
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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.
In ΔABC, R and Q are mid points of AB and CA respectively.
Proof :
∴ 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,
∠DPQ = ∠DPR + ∠RPQ = ∠A + ∠C ...(1)
∴ ∠BPR = ∠C (Corresponding angles)
RQ || BC and BR is the transversal,
To prove : P, Q, R and D are concyclic.
In ΔABC, R and Q are mid points of AB and CA respectively.
Proof :
∴ 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,
∠DPQ = ∠DPR + ∠RPQ = ∠A + ∠C ...(1)
∴ ∠BPR = ∠C (Corresponding angles)
RQ || BC and BR is the transversal,
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