so the same valueFor an equilateral triangle with vertices at (0,0), (1,0) and (1/2,1/2*sqrt(3)) then (X,Y)=(1/2,1/(2*sqrt(3))) is the point that minimizes the sum of the distances

if you look at (d/dx) and (d/dy) at that point, and the second derivatives, it's a minimum

For this case I get (1/2, 0.288674) using derivative-free Differential Evolution and the constraint is the triangle's bounding box.