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.
Contents
[hide]Sharygin’s proof
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
Claim 3
Let be the base of height Let Prove that points and are concyclic.
Proof
tangent to
Denote Point lies on radical axis of circles centered at and with the radii and respectively. Therefore points and are concyclic.
Claim 4
Prove that points and are concyclic.
Proof
and are concyclic points and are concyclic.
Sharygin’s proof
The incircle and the nine-point circle of a triangle are tangent to each other.
Proof
Let
According claim 4, each of this point lyes on
and have not more then two common point, so two of points and are coincide.
Therefore these two points coincide with point witch means that
is the center of similarity of and therefore there is no second point of intersection of and
We conclude that these circles are tangent to each other at point
vladimir.shelomovskii@gmail.com, vvsss
Scalene triangle with angle 60^\circ
The Feuerbach point of a scalene triangle lies on one of its bisectors. Prove that the angle corresponding to the bisector is
Proof
Denote given triangle, semiperimeter of are incircle, B-excircle, and nine points circle centered at and Feuerbach point, the inradius, exradius.
It is known that has diameter (half of diameter of circumcircle).
is tangent to points and lies on bisector, so Another proof Feuerbach point of a scalene triangle .
vladimir.shelomovskii@gmail.com, vvsss