Difference between revisions of "Isogonal conjugate"
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==Circumcircle of pedal triangles== | ==Circumcircle of pedal triangles== | ||
+ | [[File:Common circle.png|380px|right]] | ||
Let <math>Q</math> be the isogonal conjugate of a point <math>P</math> with respect to a triangle <math>\triangle ABC.</math> | Let <math>Q</math> be the isogonal conjugate of a point <math>P</math> with respect to a triangle <math>\triangle ABC.</math> | ||
Let <math>E, D, F</math> be the projection <math>P</math> on sides <math>AC, BC, AB,</math> respectively. | Let <math>E, D, F</math> be the projection <math>P</math> on sides <math>AC, BC, AB,</math> respectively. | ||
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Let <math>\theta = \angle ACP = \angle BCQ, \Theta = \angle ACQ = \angle BCP.</math> | Let <math>\theta = \angle ACP = \angle BCQ, \Theta = \angle ACQ = \angle BCP.</math> | ||
− | <math>CE \cdot CE' = PC | + | <math>CE \cdot CE' = PC \cos \theta \cdot QC \cos \Theta = PC \cos \Theta \cdot QC \cos \theta = CD \cdot CD'.</math> |
+ | Hence points <math>D, D', E, E'</math> are concyclic. | ||
Similarly points <math>D, D', F, F'</math> are concyclic and points <math>F, F', E, E'</math> are concyclic. | Similarly points <math>D, D', F, F'</math> are concyclic and points <math>F, F', E, E'</math> are concyclic. |
Revision as of 11:58, 6 February 2023
Isogonal conjugates are pairs of points in the plane with respect to a certain triangle.
Contents
[hide]Definition of isogonal conjugate of a point
Let be a point in the plane, and let
be a triangle. We will denote by
the lines
. Let
denote the lines
,
,
, respectively. Let
,
,
be the reflections of
,
,
over the angle bisectors of angles
,
,
, respectively. Then lines
,
,
concur at a point
, called the isogonal conjugate of
with respect to triangle
.
Proof
By our constructions of the lines ,
, and this statement remains true after permuting
. Therefore by the trigonometric form of Ceva's Theorem
so again by the trigonometric form of Ceva, the lines
concur, as was to be proven.
Second definition
Let triangle be given. Let point
lies in the plane of
Let the reflections of
in the sidelines
be
Then the circumcenter of the
is the isogonal conjugate of
Proof
common
Similarly
is the circumcenter of the
Let point be the point with barycentric coordinates
Then
has barycentric coordinates
vladimir.shelomovskii@gmail.com, vvsss
Distance to the sides of the triangle
Let be the isogonal conjugate of a point
with respect to a triangle
Let and
be the projection
on sides
and
respectively.
Let and
be the projection
on sides
and
respectively.
Then
Proof
Let
vladimir.shelomovskii@gmail.com, vvsss
Circumcircle of pedal triangles
Let be the isogonal conjugate of a point
with respect to a triangle
Let
be the projection
on sides
respectively.
Let be the projection
on sides
respectively.
Then points are concyclic.
Proof
Let
Hence points
are concyclic.
Similarly points are concyclic and points
are concyclic.
Therefore points are concyclic.
vladimir.shelomovskii@gmail.com, vvsss
Problems
Olympiad
Given a nonisosceles, nonright triangle let
denote the center of its circumscribed circle, and let
and
be the midpoints of sides
and
respectively. Point
is located on the ray
so that
is similar to
. Points
and
on rays
and
respectively, are defined similarly. Prove that lines
and
are concurrent, i.e. these three lines intersect at a point. (Source)
Let be a given point inside quadrilateral
. Points
and
are located within
such that
,
,
,
. Prove that
if and only if
. (Source)