Question 18
A fixed circle $(O,R)$ and a fixed point $A$ on it are given. Points $B,C$ vary on the circle with $\angle BAC=\alpha$ constant. On the rays $BA,CA$ take $E,F$, respectively, such that $BE=CF=BC$.
a) If $r$ is the radius of the circle $(AEF)$, prove that
$$r\ge R\frac{\sin((\pi-3\alpha)/4)}{\sin((\pi+\alpha)/4)}.$$
b) Let $P$ be the midpoint of the arc $AC$ not containing $B$, and $Q$ the midpoint of the arc $AB$ not containing $C$. The line through $O$ perpendicular to $EF$ meets the lines $AB,AC$ at $M,N$, respectively. Prove that $PM\cap QN$ always belongs to a fixed line.
a) If $r$ is the radius of the circle $(AEF)$, prove that
$$r\ge R\frac{\sin((\pi-3\alpha)/4)}{\sin((\pi+\alpha)/4)}.$$
b) Let $P$ be the midpoint of the arc $AC$ not containing $B$, and $Q$ the midpoint of the arc $AB$ not containing $C$. The line through $O$ perpendicular to $EF$ meets the lines $AB,AC$ at $M,N$, respectively. Prove that $PM\cap QN$ always belongs to a fixed line.