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 | + | ==Sharygin’s proof== |
<math>1998, 24^{th}</math> Russian math olympiad | <math>1998, 24^{th}</math> Russian math olympiad | ||
[[File:Feuerbach 1.png|500px|right]] | [[File:Feuerbach 1.png|500px|right]] | ||
− | + | ===Claim 1=== | |
− | |||
Let <math>D</math> be the base of the bisector of angle A of scalene triangle <math>\triangle ABC.</math> | Let <math>D</math> be the base of the bisector of angle A of scalene triangle <math>\triangle ABC.</math> | ||
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<math>AE, BE', CE''</math> are concurrent at the homothetic center of <math>\triangle ABC</math> and <math>\triangle EE'E''.</math> | <math>AE, BE', CE''</math> are concurrent at the homothetic center of <math>\triangle ABC</math> and <math>\triangle EE'E''.</math> | ||
− | + | ===Claim 2=== | |
[[File:Feuerbach 2.png|500px|right]] | [[File:Feuerbach 2.png|500px|right]] | ||
Let <math>M, M',</math> and <math>M''</math> be the midpoints <math>BC, AC,</math> and <math>AB,</math> respectively. Points <math>E, E',</math> and <math>E''</math> was defined at Claim 1. | Let <math>M, M',</math> and <math>M''</math> be the midpoints <math>BC, AC,</math> and <math>AB,</math> respectively. Points <math>E, E',</math> and <math>E''</math> was defined at Claim 1. | ||
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Therefore <math>MH \cdot MD = MT^2 = ME \cdot MF \implies</math> points <math>F, E, D,</math> and <math>H</math> are concyclic. | Therefore <math>MH \cdot MD = MT^2 = ME \cdot MF \implies</math> points <math>F, E, D,</math> and <math>H</math> are concyclic. | ||
+ | ===Claim 3=== | ||
+ | [[File:Feuerbach 3.png|500px|right]] | ||
+ | Let <math>H</math> be the base of height <math>AH.</math> Let <math>F_0 = ME \cap \omega \ne E.</math> | ||
+ | Prove that points <math>F_0, E, D,</math> and <math>H</math> are concyclic. | ||
+ | |||
+ | <i><b>Proof</b></i> | ||
+ | |||
+ | <math>MT</math> tangent to <math>\omega \implies MT^2 = ME \cdot MF_0.</math> | ||
+ | Denote <math>a = BC, b = AC, c = AB.</math> | ||
+ | <cmath>BD = \frac {ac}{b+c}, BM = \frac {a}{2} \implies MD = \frac {a(b-c)}{2(b+c)}.</cmath> | ||
+ | <cmath>BT = \frac {a+c-b}{2} \implies MT = \frac {b-c}{2}.</cmath> | ||
+ | Point <math>H</math> lies on radical axis of circles centered at <math>B</math> and <math>C</math> with the radii <math>c</math> and <math>b,</math> respectively. So <math>BH = \frac {a}{2} - \frac {b^2 – c^2}{2a} \implies HM = \frac {b^2 – c^2}{2a}.</math> | ||
+ | Therefore <math>MH \cdot MD = MT^2 = ME \cdot MF_0 \implies</math> points <math>F_0, E, D,</math> and <math>H</math> are concyclic. |
Revision as of 08:32, 29 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 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
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. So
Therefore points
and
are concyclic.
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. So
Therefore
points
and
are concyclic.