Question

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

Answer 1

Given: Two circles with centers $A$ and $B$, which intersect each other at $C$ and $D$.

To prove : $\angle ACB=\angle ADB$

Construction : Join $AC,AD,BD$ and $BC$.

Proof : In triangles $ACB$ and $ADB$, we have

$AC=AD$ ...Radii of the same circle

$BC=BD$ ...Radii of the same circle

$AB=AB$ ...Common

Therefore, by SSS criterion of congruence,

$△ACB\cong △ADB$

$\Rightarrow \angle ACB=\angle ADB$ ...CPCT