Isogonal conjugate
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
Points and
have not isogonal conjugate points.
Another points of sidelines have points
respectively as isogonal conjugate points.
Proof
common
Similarly
is the circumcenter of the
From definition 1 we get that is the isogonal conjugate of
It is clear that each point has the unique isogonal conjugate point.
Let point be the point with barycentric coordinates
Then
has barycentric coordinates
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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
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Sign of isogonally conjugate points
Let triangle and points
and
inside it be given.
Let be the projections
on sides
respectively.
Let be the projections
on sides
respectively.
Let Prove that point
is the isogonal conjugate of a point
with respect to a triangle
One can prove similar theorem in the case outside
Proof
Denote
Similarly
point
is the isogonal conjugate of a point
with respect to a triangle
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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.
The midpoint is circumcenter of
Proof
Let
Hence points
are concyclic.
is trapezoid,
the midpoint is circumcenter of
Similarly points are concyclic and points
are concyclic.
Therefore points are concyclic, so the midpoint
is circumcenter of
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Circles
Let be the isogonal conjugate of a point
with respect to a triangle
Let
be the circumcenter of
Let
be the circumcenter of
Prove that points
and
are inverses with respect to the circumcircle of
Proof
The circumcenter of point
and points
and
lies on the perpendicular bisector of
Similarly
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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)