I am reading about billiards in an ellipse and am a little stuck on a couple of points in the following proof (see image).
My main doubt is the highlighted line at the bottom: "Notice that line $l$ bisects $\angle F_2 A F'_2$." I cannot quite understand this. Does the highlighted line before that imply that line $m$ is a normal to the tangent? Then it makes sense.
My idea was that if we join the line segment $F'_1 F'_2$, then $\triangle F'_1 A F'_2 \cong \triangle F_1 A F_2$. So, if we fold the outside triangle onto the triangle inside the ellipse, the crease on the paper would be the line $l$, right? I think the first highlighted line implies the second. But I cannot quite prove this in writing. Can somebody please help?
Thank you.
Let's prove that normal $t$ to an angle bisector of $\angle ACB$ is tangent to ellipse with focuses at $A$ and $B$ at $C$.
Proof: Let $B'$ be a reflection over $t$ and suppose that $t$ shares another point $D\ne C$ with ellipse. Then we have $$AD +DB' = AD+DB =AC+BC = AC+B'C = AB'$$ But this contradicts triangle inequality in $AB'D$ and we are done.
