Minimum Distance Problem with several points

Question

Here is Heron's Shortest Distance Problem:

"Given two points $A$ and $B$ on one side of a line, find $C$ a point on the straight line, that minimizes $AC+BC$".

The answer is solved by the symmetry technique(See Heron's shortest distance problem).

I am interested in extending this problem as follows: "Given $n$ points $A_1,...,A_n$ on one side of a line, find $C$ a point on the x-axis, that minimizes $A_1C+A_2C+...+A_nC$".

Is there a geometric approach to answer this question? Even for three points is very interesting. Note that using derivatives needs lots of calculations.

Answer 1

If a geometric solution would require a straightedge-and-compass construction, then the answer is no.

Consider the simple case of $A_1=A_2=(0,1)$, $A_3=(1,\sqrt{2})$. Then the exact minimum for $C$ occurs at $(x,0)$ where we or WolframAlpha can calculate

The cube roots show that this is not a constructible number, and in that sense there is no geometric solution to the problem.