Question 161
Let $\triangle ABC$ be an acute triangle, and let $K$ be a point on the extension ray of $BC$ (so that $C$ lies on the segment $BK$). Let $P$ be a point such that $BP = BK$ and $PK \parallel AB$, and let $Q$ be a point such that $CQ = CK$ and $QK \parallel AC$. Assume that the circumcircle of $\triangle PQK$ and the line $AK$ intersects at another point $T$.
(1) Prove that $\angle APB + \angle BTC = \angle CQA$.
(2) Prove that $AP \cdot BT \cdot CQ = AQ \cdot BP \cdot CT$.
(1) Prove that $\angle APB + \angle BTC = \angle CQA$.
(2) Prove that $AP \cdot BT \cdot CQ = AQ \cdot BP \cdot CT$.