The point F at which the incircle and nine-point circle are tangent. It has triangle center function α = 1 - cos(B - C) and is Kimberling center X_11. If F is the Feuerbach point a triangle Δ A B C and X, Y, and Z are the midpoints of the sides B C, C A, and A B, respectively, then one of the distances left bracketing bar F X right bracketing bar , left bracketing bar F Y right bracketing bar , and left bracketing bar F Z right bracketing bar is equal to the sum of the two others. For example, in the above figure, left bracketing bar F X right bracketing bar = left bracketing bar F Y right bracketing bar + left bracketing bar F Z right bracketing bar .

Feuerbach antipode | Feuerbach's theorem | Feuerbach triangle | incircle | nine-point circle

Karl Wilhelm Feuerbach