I'm trying to follow this proof for reflection property of ellipse, I understood the idea of reflection and using it to minimize distances because it's the same concept which we use to solve heron's problem.
However, I don't get the second part of his arguments where they dtldarek actually speaks of the ellipse. How did he argue that the point on tangent is the one which minimizes the "heron's distance" between F' and F?
Note: By heron's distance I mean shortest distance with two points on the constraint that you must first move from one of the point to a line and then from the line you must move to theother point
By definition, all points $Q$ on the tangent, different from tangency point $T$, are external to the ellipse. Hence $FQ+F'Q>FT+F'T$, that is $T$ minimises "heron's distance".