Question

Two congurent circles intersect each other at points $A$ and $B$. Through $A$ any line segment $PAQ$ is drawn so that $P,Q$ lie on the two circles. Prove that $BP=BQ$.

Answer 1

Let point $C$ and $D$ be the centres of the two congruent circles.

Let $AB$ be the common chord.

$\therefore arcAEB=arcAFB$ ....arcs of congruent chords of congruent circles

$\therefore \angle APB=\angle AQB$ ....angles subtended by congruent arcs ....(1)

In $△BPQ$,

$\angle APB=\angle AQB$ ...from (1)

i.e. $\angle QPB=\angle PQB$

$\therefore BQ=BP$ ...converse of isosceles triangle theorem