Difference between revisions of "Feuerbach point"
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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. | 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== | ==Sharygin’s prove== | ||
| − | 1998, | + | <math>1998, 24^{th}</math> Russian math olympiad |
| − | + | [[File:Feuerbach 1.png|500px|right]] | |
<i><b>Claim 1</b></i> | <i><b>Claim 1</b></i> | ||
Revision as of 11:16, 27 December 2023
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, 
![\[\angle TIT'' = 180^\circ - 2 \beta, \angle ADB = 180^\circ - \alpha - 2 \beta,\]](http://latex.artofproblemsolving.com/4/1/0/410643b4d81dd3998b9efa44894cb8f6efeec5c8.png)
![\[\angle DIT = 90^\circ - \angle ADB = \alpha + 2 \beta – 90^\circ = \beta -\gamma, \angle EID = \angle TID \implies\]](http://latex.artofproblemsolving.com/4/a/5/4a5a370aeca93460ff302ce5472455431ebdfacc.png)
![\[\angle T''IE = \angle T''IT + 2 \angle TID = 180^\circ - 2 \beta + 2(\beta - \gamma) = 180^\circ - 2 \gamma.\]](http://latex.artofproblemsolving.com/b/1/3/b134bf4881227acfc2376a6e13868673ed84d390.png)
Similarly, 
points 
and 
are symmetric with respect 
Similarly, 
are concurrent at the homothetic center of 
and 
