Question

Question 1
Prove that line of centres of two intersecting circles subtends equal angles at the two points of intersection.

Answer 1

Let two circles having their centres as O and O' intersect each other at point A and B respectively. Let us join OO'.

In $\Delta AO{O}^{\prime }andBO{O}^{\prime }$,
OA = OB (Radius of circle with centre O)
O'A = O'B (Radius of circle with centre O')
OO' = OO'(Common)
$\Delta AO{O}^{\prime }\cong \Delta BO{O}^{\prime }$ (By SSS congruence rule)
$\angle OA{O}^{\prime }=\angle OB{O}^{\prime }\left(ByCPCT\right)$
Therefore, line of centres of two intersecting circles subtends equal angles at the two points of intersection.