There are two theorems commonly known as Feuerbach's theorem. The first states that circle which passes through the feet of the perpendiculars
dropped from the polygon vertices of any triangle
on the sides opposite them passes also through the midpoints
of these sides as well as through the midpoint of the
segments which join the polygon vertices to the
point of intersection of the perpendicular. Such
a circle is called a nine-point circle.
The proposition most frequently called Feuerbach's theorem states that the nine-point circle of any triangle is tangent
internally to the incircle and tangent
externally to the three excircles. This theorem was
first published by Feuerbach (1822). Many proofs have been given (Elder 1960), with
the simplest being the one presented by M'Clelland (1891, p. 225) and Lachlan
(1893, p. 74).