Isogonal conjugate
Isogonal conjugates are pairs of points in the plane with respect to a certain triangle.
Contents
- 1 The isogonal theorem
- 2 Perpendicularity
- 3 Fixed point
- 4 Bisector
- 5 Isogonal of the diagonal of a quadrilateral
- 6 Isogonals in trapezium
- 7 Isogonal of the bisector of the triangle
- 8 Definition of isogonal conjugate of a point
- 9 Second definition
- 10 Distance to the sides of the triangle
- 11 Sign of isogonally conjugate points
- 12 Circumcircle of pedal triangles
- 13 Common circumcircle of the pedal triangles as the sign of isogonally conjugate points
- 14 Circles
- 15 Problems
The isogonal theorem
Isogonal lines definition
Let a line and a point
lying on
be given. A pair of lines symmetric with respect to
and containing the point
be called isogonals with respect to the pair
Sometimes it is convenient to take one pair of isogonals as the base one, for example, and
are the base pair. Then we call the remaining pairs as isogonals with respect to the angle
Projective transformation
It is known that the transformation that maps a point with coordinates into a point with coordinates
is projective.
If the abscissa axis coincides with the line and the origin coincides with the point
then the isogonals define the equations
and the lines
symmetrical with respect to the line
become their images.
It is clear that, under the reverse transformation (also projective), such pairs of lines become isogonals, and the points equidistant from lie on the isogonals.
The isogonal theorem
Let two pairs of isogonals and
be given. Let lines
and
intersect at point
Let lines
and
intersect at point
Prove that
and
are the isogonals with respect to the pair
Proof
Let us perform a projective transformation of the plane that maps the point into a point at infinity and the line
maps to itself. In this case, the isogonals turn into a pair of straight lines parallel to
and equidistant from
The reverse (also projective) transformation maps the points equidistant from onto isogonals.
Let the images of isogonals are vertical lines. Let coordinates of images of points be
Equation of a straight line
is
Equation of a straight line is
Point abscissa
Equation of a straight line is
Equation of a straight line is
Point abscissa
Preimages of the points and
lie on the isogonals.
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Perpendicularity
Let triangle be given. Right triangles
and
with hypotenuses
and
are constructed on sides
and
to the outer (inner) side of
Let
Prove that
Proof
Let be the bisector of
and
are isogonals with respect to the pair
and
are isogonals with respect to the pair
and
are isogonals with respect to the pair
in accordance with The isogonal theorem.
is diameter of circumcircle of
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Fixed point
Let fixed triangle be given. Let points
and
on sidelines
and
respectively be the arbitrary points.
Let be the point on sideline
such that
Prove that line pass through the fixed point.
Proof
We will prove that point symmetric
with respect
lies on
.
and
are isogonals with respect to
points
and
lie on isogonals with respect to
in accordance with The isogonal theorem.
Point symmetric
with respect
lies on isogonal
with respect to
that is
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Bisector
Let a convex quadrilateral be given. Let
and
be the incenters of triangles
and
respectively. Let
and
be the A-excenters of triangles
and
respectively.
Prove that is the bisector of
Proof
and
are isogonals with respect to the angle
and
are isogonals with respect to the angle
in accordance with The isogonal theorem.
Denote
WLOG,
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Isogonal of the diagonal of a quadrilateral
Given a quadrilateral and a point
on its diagonal such that
Let
Prove that
Proof
Let us perform a projective transformation of the plane that maps the point to a point at infinity and the line
into itself.
In this case, the images of points and
are equidistant from the image of
the point (midpoint of
lies on
contains the midpoints of
and
is the Gauss line of the complete quadrilateral
bisects
the preimages of the points and
lie on the isogonals
and
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Isogonals in trapezium
Let the trapezoid be given. Denote
The point on the smaller base
is such that
Prove that
Proof
Therefore
and
are isogonals.
Let us perform a projective transformation of the plane that maps the point to a point at infinity and the line
into itself.
In this case, the images of points and
are equidistant from the image of
contains the midpoints of
and
, that is,
is the Gauss line of the complete quadrilateral
bisects
The preimages of the points and
lie on the isogonals
and
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Isogonal of the bisector of the triangle
The triangle be given. The point
chosen on the bisector
Denote
Prove that
Proof
Let us perform a projective transformation of the plane that maps the point to a point at infinity and the line
into itself.
In this case, the images of segments and
are equidistant from the image of
is midpoint of
and midpoint
is parallelogramm
distances from
and
to
are equal
Preimages
and
are isogonals with respect
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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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Common circumcircle of the pedal triangles as the 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 points be concyclic and none of them lies on the sidelines of
Then point is the isogonal conjugate of a point
with respect to a triangle
This follows from the uniqueness of the conjugate point and the fact that the line intersects the circle in at most two points.
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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)