Why are these angles equal?

Question

-see picture above- (from Billiards and Geometry by Serge Tabachnikov)

I don't understand why the angles $F_2BA_1$ and $F_1BA_0$ are equal (I do understand the conclusion, that follows from the optical property of the ellipse). Can somebody help me out?

Answer 1

$\angle F_2BA_1=\angle F_1'BA_0$ (vertical angles);

$\angle F_1'BA_0=\angle F_1BA_0$ (corresponding angles in the reflection about $A_0A_1$).

Hence, by transitivity, $\angle F_2BA_1=\angle F_1BA_0$.