Difference between revisions of "Steiner line"
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Usually the point <math>P</math> is called the anti-Steiner point of the <math>H-line</math> with respect to <math>\triangle ABC.</math> | Usually the point <math>P</math> is called the anti-Steiner point of the <math>H-line</math> with respect to <math>\triangle ABC.</math> | ||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ==Ortholine== | ||
+ | [[File:Ortholine.png|500px|right]] | ||
+ | Let four lines made four triangles of a complete quadrilateral. | ||
+ | |||
+ | In the diagram these are <math>\triangle ABC, \triangle ADE, \triangle CEF, \triangle BDF.</math> | ||
+ | |||
+ | Let points <math>H, H_A, H_B,</math> and <math>H_C</math> be the orthocenters of <math>\triangle ABC, \triangle ADE, \triangle BDF,</math> and <math>\triangle CEF,</math> respectively. | ||
+ | |||
+ | Prove that points <math>H, H_A, H_B,</math> and <math>H_C</math> are collinear. | ||
+ | |||
+ | <i><b>Proof</b></i> | ||
+ | |||
+ | Let <math>M</math> be Miquel point of a complete quadrilateral. | ||
+ | |||
+ | Line <math>KLMN</math> is the line which contain <math>4</math> Simson lines of <math>4</math> triangles. | ||
+ | |||
+ | Using homothety centered at <math>M</math> with ratio <math>2</math> we get <math>4</math> coinciding Stainer lines which contain points <math>H, H_A, H_B,</math> and <math>H_C</math>. | ||
+ | *[[Miquel’s point]] | ||
+ | *[[Simson line]] | ||
'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' |
Revision as of 15:41, 7 December 2022
Steiner line
Let be a triangle with orthocenter
is a point on the circumcircle
of
Let and
be the reflections of
in three lines which contains edges
and
respectively.
Prove that and
are collinear. Respective line is known as the Steiner line of point
with respect to
Proof
Let and
be the foots of the perpendiculars dropped from
to lines
and
respectively.
WLOG, Steiner line cross at
and
at
The line is Simson line of point
with respect of
is midpoint of segment
homothety centered at
with ratio
sends point
to a point
Similarly, this homothety sends point to a point
, point
to a point
therefore this homothety send Simson line to line
Let
is simmetric to
Quadrungle is cyclic
at point
Similarly, line
at
According the Collins Claim is
therefore
vladimir.shelomovskii@gmail.com, vvsss
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
Usually the point is called the anti-Steiner point of the
with respect to
vladimir.shelomovskii@gmail.com, vvsss
Ortholine
Let four lines made four triangles of a complete quadrilateral.
In the diagram these are
Let points and
be the orthocenters of
and
respectively.
Prove that points and
are collinear.
Proof
Let be Miquel point of a complete quadrilateral.
Line is the line which contain
Simson lines of
triangles.
Using homothety centered at with ratio
we get
coinciding Stainer lines which contain points
and
.
vladimir.shelomovskii@gmail.com, vvsss