Question 54
For points $P$, $Q$, $R$, let $E_{P, Q}(R)$ denote the ellipse with foci $P$ and $Q$ through $R$.
Points $T$, $A$, $B$, $C$ are points on a line in that order. A ray from $A$ intersects $E_{A, B}(T)$ at $Q$ and $E_{A, C}(T)$ at $P$. Show that $E_{B, P}(A)$ intersects $E_{C, Q}(A)$ at two points.
Points $T$, $A$, $B$, $C$ are points on a line in that order. A ray from $A$ intersects $E_{A, B}(T)$ at $Q$ and $E_{A, C}(T)$ at $P$. Show that $E_{B, P}(A)$ intersects $E_{C, Q}(A)$ at two points.