- Source: Circumcevian triangle
In Euclidean geometry, a circumcevian triangle is a special triangle associated with a reference triangle and a point in the plane of the triangle. It is also associated with the circumcircle of the reference triangle.
Definition
Let P be a point in the plane of the reference triangle △ABC. Let the lines AP, BP, CP intersect the circumcircle of △ABC at A', B', C'. The triangle △A'B'C' is called the circumcevian triangle of P with reference to △ABC.
Coordinates
Let a,b,c be the side lengths of triangle △ABC and let the trilinear coordinates of P be α : β : γ. Then the trilinear coordinates of the vertices of the circumcevian triangle of P are as follows:
A
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a
β
γ
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(
b
γ
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c
β
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β
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b
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c
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γ
B
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c
α
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a
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α
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b
γ
α
:
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c
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a
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γ
C
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a
β
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b
α
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α
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a
β
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b
α
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β
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−
c
α
β
{\displaystyle {\begin{array}{rccccc}A'=&-a\beta \gamma &:&(b\gamma +c\beta )\beta &:&(b\gamma +c\beta )\gamma \\B'=&(c\alpha +a\gamma )\alpha &:&-b\gamma \alpha &:&(c\alpha +a\gamma )\gamma \\C'=&(a\beta +b\alpha )\alpha &:&(a\beta +b\alpha )\beta &:&-c\alpha \beta \end{array}}}
Some properties
Every triangle inscribed in the circumcircle of the reference triangle ABC is congruent to exactly one circumcevian triangle.
The circumcevian triangle of P is similar to the pedal triangle of P.
The McCay cubic is the locus of point P such that the circumcevian triangle of P and ABC are orthologic.
See also
Cevian
Ceva's theorem