Question

Two circles intersect at A and B. From a point P on one of these circles, two line segments PAC and PBD are drawn, intersecting the other circle at C and D respectively. Prove the CD is parallel to the tangent at P.

Answer 1

Join AB and let XY be the tangent at P. Then by alternate segment theorem,

$\angle APX=\angle ABP$ ……………(i)

Next, ABCD is a cyclic quadrilateral, therefore, by the theorem sum of the opposite angles of a quadrilateral is 180^{\circ}

$\angle ABD+\angle ACD={180}^{\circ }$

Also, $\angle ABD=\angle ABP={180}^{\circ }$ (Linear Pair)

$\therefore \angle ACD=\angle ABP$ ...........(ii)

From (i) and (ii),

$\angle ACD=\angle APX$

$\therefore XY\parallel CD$ (Since alternate angles are equal).