Two circles k1 and k2 intersect in the points A and B. A first straight line through B intersects k1 in C and k2 in E. A second straight line through B intersects k1 in D and k2 in F. B shall be between C and E. B shall also be between D and F. Finally, let M be the midpoint of CE and N the midpoint of DF.

Prove that the triangles ACD, AEF and AMN are similar.

https://www.cut-the-knot.org/proofs/longest.shtml.

To solve your problem, one has only to consider the difference of two equal angles at, say, D and E.