Feuerbach point

It is listed as X(11) in Clark Kimberling's Encyclopedia of Triangle Centers, and is named after Karl Wilhelm Feuerbach.

[1][2] Feuerbach's theorem, published by Feuerbach in 1822,[3] states more generally that the nine-point circle is tangent to the three excircles of the triangle as well as its incircle.

The nine-point circle passes through these three midpoints; thus, it is the circumcircle of the medial triangle.

These two circles meet in a single point, where they are tangent to each other.

These are circles that are each tangent to the three lines through the triangle's sides.

Each excircle touches one of these lines from the opposite side of the triangle, and is on the same side as the triangle for the other two lines.

Like the incircle, the excircles are all tangent to the nine-point circle.

The Feuerbach point lies on the line through the centers of the two tangent circles that define it.

Then,[7][8] or, equivalently, the largest of the three distances equals the sum of the other two.

where O is the reference triangle's circumcenter and I is its incenter.[8]: Propos.

3 The latter property also holds for the tangency point of any of the excircles with the nine–point circle: the greatest distance from this tangency to one of the original triangle's side midpoints equals the sum of the distances to the other two side midpoints.

[8] If the incircle of triangle ABC touches the sides BC, CA, AB at X, Y, and Z respectively, and the midpoints of these sides are respectively P, Q, and R, then with Feuerbach point F the triangles FPX, FQY, and FRZ are similar to the triangles AOI, BOI, COI respectively.[8]: Propos.