Question 1
Let $ABCD$ be a convex quadrilateral inscribed in a circle with centre $O$ and radius $R$, with neither pair of opposite sides parallel. Let $E=AC\cap BD$. The internal bisector of $\angle AEB$ meets the lines $AB,BC,CD,DA$ at $M,N,P,Q$, respectively.
1. Prove that $(AQM),(BMN),(CNP),(DPQ)$ have exactly one common point $K$.
2. Put $m=\min\{AC,BD\}$. Prove
$$OK\le\frac{2R^2}{\sqrt{4R^2-m^2}}.$$
1. Prove that $(AQM),(BMN),(CNP),(DPQ)$ have exactly one common point $K$.
2. Put $m=\min\{AC,BD\}$. Prove
$$OK\le\frac{2R^2}{\sqrt{4R^2-m^2}}.$$