In the plane of any triangle ABC, let

A'BC = the equilateral triangle shown on side BC,
AB'C = the equilateral triangle shown on side CA,
ABC' = the equilateral triangle shown on side AB.

The lines AA', BB', CC' meet in the Fermat point, said to be the first triangle center discovered after ancient Greek times. The great French mathematician Pierre Fermat, posed as a problem the search for a point P for which the sum PA + PB + PC of the distances from P to the vertices is as small as possible. Torricelli proved that the Fermat point, labeled F in the diagram, is the solution if each angle of triangle ABC is less than 120 degrees. Sometimes, F is called the Fermat-Toricelli point.

The Fermat point is also known as the 1st isogonic center, the roots iso and gon meaning equal-angle. This is because the angles BFC, CFA, AFB are all equal. (The 2nd isogonic center is obtained using the other three equilateral triangles on the sides of triangle ABC.

Triangle Centers
Clark Kimberling Home Page