Question 150
Suppose an ellipse with points $B_0$ and $B_1$ as the foci intercepts side $AB_i$ of $\triangle AB_0B_1$ at $C_i$ ($i = 0, 1$). Taking an arbitrary point $P_0$ on the extending line of $AB_0$, draw arc $\widehat{P_0Q_0}$ with $B_0$, $B_0P_0$ as the center and radius respectively, intercepting the extending line of $C_1B_0$ at $Q_0$. Draw arc $\widehat{Q_0P_1}$ with $C_1$, $C_1Q_0$ as the center and radius respectively,
intercepting the extending line of $B_1A$ at $P_1$. Draw arc $\widehat{P_1Q_1}$ with $B_1$, $B_1P_1$ as the center and radius respectively, intercepting the extending line of $B_1C_0$ at $Q_1$. Draw arc $\widehat{Q_1P'_0}$ with $C_0$, $C_0Q_1$ as the center and radius respectively, intercepting the extending line of $AB_0$ at $P'_0$. Prove that
(1) $P'_0$ and $P_0$ are coincident, and arcs $\widehat{P_0Q_0}$ and $\widehat{P_0Q_1}$ are tangent to each other at $P_0$.
(2) Points $P_0, Q_0, Q_1, P_1$ are concyclic.
intercepting the extending line of $B_1A$ at $P_1$. Draw arc $\widehat{P_1Q_1}$ with $B_1$, $B_1P_1$ as the center and radius respectively, intercepting the extending line of $B_1C_0$ at $Q_1$. Draw arc $\widehat{Q_1P'_0}$ with $C_0$, $C_0Q_1$ as the center and radius respectively, intercepting the extending line of $AB_0$ at $P'_0$. Prove that
(1) $P'_0$ and $P_0$ are coincident, and arcs $\widehat{P_0Q_0}$ and $\widehat{P_0Q_1}$ are tangent to each other at $P_0$.
(2) Points $P_0, Q_0, Q_1, P_1$ are concyclic.