
This elegant proof comes from website http://www.ies.co.jp/math/java/conics/focus_ellipse/focus_ellipse.html
If the interior of ellipse is silvered to produce a mirror, rays originated at ellipse's focus are reflected to the other focus of the ellipse. 
Part 1. 

Part 2. There are two point A, B, a line L and a moving point Q on L. Point A' is the reflection of A about L. When AQ + BQ = minimum at Q = P, BQ and QA' form BA', which is a straight line. Angles made by AP, BP and L are equal to each other. 
http://www.ies.co.jp/math/java/conics/focus_ellipse/focus_ellipse.html
The webmaster and author of this Math Help site is Graeme McRae.