Feuerbach's Theorem

The nine-point circle of a given triangle contains the midpoint of each side of the triangle, the foot of each altitude and the midpoint of the line segment from each vertex of the triangle to the orthocenter.

The nine-point circle is also called Euler's circle or the Feuerbach circle.

Feuerbach's Theorem states that the nine-point circle and the incircle of the triangle are mutually tangent (and touch at the Feuerbach point).



Feuerbach also proved the ninepoint circle was tangent to the three ex-circles of the triangle.