Question

Two circles intersect each other in points M and N. An arbitrary point A of the first circle, which is not M or N, is connected with M and N, and the straight lines AM and AN intersect the second circle again in the points B and C. Prove that the tangent to the first circle at A is parallel to the straight line BC.

Answer 1

Let AT be the tangent to the first circle at A. Then, 

∠TAM = ∠ANM (Angles in alternate segment are equal) 

⇒ ∠ANM = ∠MBC (ext. ∠ = int. opp. ∠ in a cyclic quad.) 

we have ∠TAB = ∠ABC ⇒ AT || BC. (alt. ∠s)