In the question "Minimum Distance Problem with several points", the OP has asked for a geometric solution of a generalization of Heron's problem (for $n$ points). I am interested in the (much more modest) special case of three points:
"Given three points $A$, $B$ and $C$ on the same side of a straight line, find $X$, a point on the straight line, such that it minimizes $AX+BX+CX$."
Since with two points one can solve it geometrically by (among many other methods) thinking of an ellipse expanding until it just touches the straight line, I thought that maybe one can solve the three points case by thinking on some other expanding shape (the $n$-ellipse?).
P.S. https://www.cut-the-knot.org/Curriculum/Geometry/HeronsProblem.shtml