Question

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

Answer 1

Let us take two circles with center $O$ and $O’$ such that they intersect each other at $P$ and $Q$.

Join $OP,OQ,O\text{'}P,O\text{'}Q$ and $OO\text{'}$

In $△OPO\text{'}$ and $△OQO\text{'}$

$OP=OQ$ [ Radii of the same circle]

$PO\text{'}=QO\text{'}$ [ Radii of the same circle]

$OO\text{'}=OO\text{'}$ [ Common side of the triangle]

By SSS congruence criteria, $△OPO\text{'}\cong △OQO\text{'}$

$\Rightarrow \angle OPO\text{'}=\angle OQO\text{'}$ [ By CPCT]

Hence, proved that the line joining the centers of two intersecting circles subtends equal angles at the two points of intersection.