Question 51
Let $ABCD$ be a quadrilateral inscribed in circle $(O, R)$ and is not a trapezoid. Two lines $AC$, $BD$ intersect each other at $E$ and the bisector of $\angle AEB$ meets the lines $AB$, $BC$, $CD$, $DA$ at $M$, $N$, $P$, $Q$, respectively.
1. Prove that four lines $(AQM)$, $(BMN)$, $(CNP)$, $(DPQ)$ concur at a unique point, name it as $K$.
2. Let $\min\{AC, BD\} = m$. Prove that $\mathrm{OK} \le \frac{2R^2}{\sqrt{4R^2 - m^2}}$.
1. Prove that four lines $(AQM)$, $(BMN)$, $(CNP)$, $(DPQ)$ concur at a unique point, name it as $K$.
2. Let $\min\{AC, BD\} = m$. Prove that $\mathrm{OK} \le \frac{2R^2}{\sqrt{4R^2 - m^2}}$.