Prove that line of centres of two intersecting circles subtends equal angles at the two points of intersection.

$\mathrm{AC}$and $\mathrm{BD}$are chords of a circle which bisect each other. Prove that $\mathrm{ABCD}$ is a rectangle.

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