Light reflects off a curved surface by forming equal (incidence and reflection) angles with the tangent at the point it falls on the surface. So locally a surface (curve) behaves like a flat plane (straight line). In the latter case of a straight line, the optical property has an expression in Heron's theorem, which finds a point on a line the sum of whose distances to two given points (on the same side from the line) is minimum. Heron's theorem helps explain the reflective property of ellipse.
which is the sum of distances from the two points to a point M. By definition, ellipse is a level curve of that function, i.e., a curve where the function is constant: f(M) = const. The level curves of f(M) split the plane into a family of confocal (i.e. sharing the same foci) ellipses. As the ellipses expand the function grows.
Choose one ellipse from the family, a point T on it and the tangent to the ellipse at T. The tangent has a single point in common with the selected ellipse. All its other points are outside that ellipse, i.e. on the ellipses with the values of f(M) greater than at T: f(T) < f(M), for all M outside the selected ellipse. In particular that is true for all points of the tangent at T. But then, by Heron's theorem, the focal radii at T form equal angles with the tangent.
Other properties follow from this one. For example, denote P2 the pedal point (the foot of the perpendicular) of F2 on the tangent and R2 the reflection of F2 in the tangent. P2 is the midpoint of F2R2. Also,
