first isogonic center

The first Fermat point X (or F_1) (sometimes simply called "the Fermat point, " Torricelli point, or first isogonic center) is the point X which minimizes the sum of distances from A, B, and C in an acute triangle, left bracketing bar A X right bracketing bar + left bracketing bar B X right bracketing bar + left bracketing bar C X right bracketing bar . It has equivalent triangle center functions α_13 | = | csc(A + 1/3 π) α_13 | = | b c[c^2 a^2 + (c^2 + a^2 - b^2)^2][a^2 b^2 - (a^2 + b^2 - c^2)^2][4Δ - sqrt(3)(b^2 + c^2 - a^2)] and is Kimberling center X_13. It also arises in Napoleon's theorem.

Fermat axis | Fermat points | Napoleon's theorem | second Fermat point