Question 1
In the plane two circles $(C_1),(C_2)$ intersect at $P,Q$. Their common tangent nearer $P$ than $Q$ touches $(C_1)$ at $A$ and $(C_2)$ at $B$. The tangents to $(C_1),(C_2)$ at $P$ meet the other circle again at $E,F$, respectively. Let $H,K$ be points on the lines $AF,BE$, respectively, such that $AH=AP$ and $BK=BP$. Prove that $A,H,Q,K,B$ lie on one circle.
Editorial warning: a full line permits two choices at a prescribed positive distance. The statement as printed does not select the choices needed by the source proof. The solution discusses this limitation rather than silently changing the stem.
Editorial warning: a full line permits two choices at a prescribed positive distance. The statement as printed does not select the choices needed by the source proof. The solution discusses this limitation rather than silently changing the stem.