Question

Two circles touch each other at A. Any line through A meets the circle at P. Prove that tangents at P and Q are parallel

Answer 1

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Two circles touch each other at A. Any line through A meets the circle at P. Prove that tangents at P and Q are parallel .

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Join AC,PQ and BD.

ACQP is a cyclic quadrilaterl.

∠CAP+∠PQC=180 ∘

....(1) pair of opposites in cyclic quadrilateral.

PQDB is a cyclic quadrilaterl.

∠PQD+∠DBP=180 ∘....(2) pair of opposites in cyclic quadrilateral.

∠PQC+∠PQD=180 ∘ .....(3) CQD is a straight line.

From (1), (2) and (3), we get,

∠CAP+∠DBP=180 ∘

∠CAB+∠DBA=180 ∘

If a traversal intersects 2 lines such that a pair of interior angles on same side of traversal is supplimentary, then the 2 lines are parallel.

∴AC ∥ BD

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