Question

Two circle Γ and ∑ lies in the plane intersect at two distinct points A and B and the centre of ∑ lies on Γ. Let points C and D be on Γ and ∑ respectively such that C,B and D are collinear. Let points E on ∑ be such that DE is parallel to AC. Show that AE=AB.

Answer 1

If O is the entre of ∑, then we have

This shows that  ∠EAB = ∠EBA and hence AE = AB.