Difference between revisions of "Steiner line"
(Created page with "==Steiner line== Let <math>ABC</math> be a triangle with orthocenter <math>H. S</math> is a point on the circumcircle <math>\Omega</math> of <math>\triangle ABC.</math> Then,...") |
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Let <math>H_A, H_B,</math> and <math>H_C</math> be the points symmetric to <math>H</math> with respect <math>BC, AC,</math> and <math>AB,</math> respectively. | Let <math>H_A, H_B,</math> and <math>H_C</math> be the points symmetric to <math>H</math> with respect <math>BC, AC,</math> and <math>AB,</math> respectively. | ||
− | Therefore <math>H_A \in l_A, H_B \in l_B, H_C \in l_C, AH = AH_B = AH_C, BH = BH_A = BH_C, CH = CH_A = CH_B \implies</ | + | Therefore <math>H_A \in l_A, H_B \in l_B, H_C \in l_C,</math> |
+ | <cmath>AH = AH_B = AH_C, BH = BH_A = BH_C, CH = CH_A = CH_B \implies</cmath> | ||
<cmath>\angle HH_BE = \angle EHH_B = \angle BHD = \angle BH_CD.</cmath> | <cmath>\angle HH_BE = \angle EHH_B = \angle BHD = \angle BH_CD.</cmath> | ||
− | Let <math>P</math> be the crosspoint of <math>l_B</math> and <math>l_C \implies BH_CH_BP</math> is cyclic <math>\implies P \in \ | + | Let <math>P</math> be the crosspoint of <math>l_B</math> and <math>l_C \implies BH_CH_BP</math> is cyclic <math>\implies P \in \Omega.</math> |
− | Similarly <math>\angle CH_BE = \angle CHE = \angle CH_A \implies CH_BH_AP</math> is cyclic <math>\implies P \in \ | + | Similarly <math>\angle CH_BE = \angle CHE = \angle CH_A \implies CH_BH_AP</math> is cyclic <math>\implies P \in \Omega \implies</math> the crosspoint of <math>l_B</math> and <math>l_A</math> is point <math>P.</math> |
'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' |
Revision as of 01:40, 7 December 2022
Steiner line
Let be a triangle with orthocenter
is a point on the circumcircle
of
Then, the reflections of
in three edges
and point
lie on a line
which is known as the Steiner line of point
with respect to
Collings Clime
Let triangle be the triangle with the orthocenter
and circumcircle
Denote
any line containing point
Let and
be the reflections of
in the edges
and
respectively.
Prove that lines and
are concurrent and the point of concurrence lies on
Proof
Let and
be the crosspoints of
with
and
respectively.
WLOG
Let
and
be the points symmetric to
with respect
and
respectively.
Therefore
Let be the crosspoint of
and
is cyclic
Similarly is cyclic
the crosspoint of
and
is point
vladimir.shelomovskii@gmail.com, vvsss