Let there be given three points $x_1, x_2, x_3$ and a line $l$ on a plane. Does there exist an explicit method of finding such a point $p$ on $l$, that sum of distances of $p$ from $x_1, x_2, x_3$ is possibly minimal? Such result for two given points and a line is trivial and known as the Heron's theorem commonly used in optics to find a path of a ray of light.
An example of three points and a point $p$ on the line
All I know about the solution is that if $p$ minimizes the sum of distances, then $$\cos\alpha+\cos\beta+\cos\gamma = 0$$
However, that is not really helpful neither for construction nor finding the solution analytically. Some rare situations may be solved using Brianchon's theorem.
That is, if we add symmetrical reflections of $x_1, x_2, x_3$ on the opposite sides of $l$ and the six points are vertices of a hexagon circumscribing a conic section, then the intersection of the main diagonals minimizes the sum of distances. However it is far from general solution.
