Two congurent...

Question

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

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Solution

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

Let $A B$ be the common chord.

$∴ a r c A E B = a r c A F B$ ....arcs of congruent chords of congruent circles

$∴ ∠ A P B = ∠ A Q B$ ....angles subtended by congruent arcs ....(1)

In $△ B P Q$ ,

$∠ A P B = ∠ A Q B$ ...from (1)

i.e. $∠ Q P B = ∠ P Q B$

$∴ B Q = B P$ ...converse of isosceles triangle theorem

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