Fermat / Torricelli point

Dependencies: Rotation properties (rotations preserve distance and angle), Triangle inequality a + b > c (a polyline is at least as long as the straight segment), Isoceles with a 60° angle ⇒ equilateral (isosceles + apex angle $60°$ $\Rightarrow$ equilateral), Linear pair sums to 180°.

Statement

Let $\triangle ABC$ have all three interior angles strictly less than $120^\circ$. Then there is a unique point $F$ inside the triangle such that for every point $P$ in the plane,

with equality if and only if $P = F$. This minimizer $F$ is called the Fermat point of $\triangle ABC$ (also the Torricelli point).

The geometric characterization of the Fermat point $F$ is: the three sides are seen at $120^\circ$ from $F$, i.e.

When some interior angle of the triangle is $\ge 120^\circ$, the minimizer degenerates to that vertex itself — this section only handles the case "all interior angles $< 120^\circ$".