$\frac{PM}{PN} = \frac{AE}{DE}$, $\frac{MN}{NP} = \frac{23}{13}$. Instead of finding $\frac{MN}{NP}$, one should prove this problem which I did: Let $ABCD$ be inscribed in a circle with center $O$. Let $E$ be the intersection point of $AC$ and $BD$, $M$ and $N$ the midpoints of the arcs $AB$ and $CD$, respectively (the arcs not containing any other vertices). Let $P$ be the intersection point of $EO$ and $MN$. Prove that $\frac{PM}{PN} = \frac{AB}{CD}$. This problem is dedicated to my teacher Le Chi De of Australia and my students Antarish Rautela, Neil Shah and David Fletcher of the United States.