A problem due to Fermat

Fermat posed the problem

Given a triangle, find the point where the sum of the distances to the vertices is a minimum.

Here is a solution

Given any triangle, construct an equilateral triangle on each side and join the vertices as shown.

These three lines meet at 60° in the so-called Fermat point.

(Actually, if the triangle has an angle ≥ 120° then this vertex gives the smallest sum.)