Question

In figure two circles intersect at two points P and Q. From a point A on a circle, two line segments APC and AQD are drawn intersecting the other circle at the points C and D. Prove that CD is parallel to the tangent at A.

Answer 1

Given: Two circles intersect at P and Q.
To prove: CD $\parallel$ tangent at A.
Proof: Join PQ. Let XY be the tangent at A. Then by alternate segment theorem,
$\angle PAX=\angle PQA$ ---- (1)
PQCD is a cylic quadrailateral, therefore, by the theorem sum of the opposite angles of the quadrilateral is ${180}^o$
$\angle PQD+\angle PQA={180}^o$ (linear pair)
$\angle PCD=\angle PQA$ ---- (2)
From (1) and (2)
$\angle PCD=\angle PAX$
Therefore, XY $\parallel$ CD (Since alternate angles are equal)