Feuerbach point
The incircle and nine-point circle of a triangle are tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is listed as X(11) in Clark Kimberling's Encyclopedia of Triangle Centers and is named after Karl Wilhelm Feuerbach.
Sharygin’s prove
Russian math olympiad
Claim 1
Let be the base of the bisector of angle A of scalene triangle
Let be a tangent different from side
to the incircle of
is the point of tangency). Similarly, we denote
and
Prove that are concurrent.
Proof
Let and
be the point of tangency of the incircle
and
and
Let
WLOG,
Similarly,
points
and
are symmetric with respect
Similarly,
are concurrent at the homothetic center of
and
Claim 2
Let and
be the midpoints
and
respectively. Points
and
was defined at Claim 1.
Prove that and
are concurrent.
Proof
are concurrent at the homothetic center of
and