Circumcevian_triangle Knowpia

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 edit

Reference triangle

△ ABC

Point

P

Circumcircle of

△ ABC

; lines between the vertices of

△ ABC

and

P

Circumcevian triangle

△ A'B'C'

of

P

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 $△ ABC$ .^[1]

Coordinates edit

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:^[2]

{\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 edit

Every triangle inscribed in the circumcircle of the reference triangle ABC is congruent to exactly one circumcevian triangle.^[2]
The circumcevian triangle of P is similar to the pedal triangle of P.^[2]
The McCay cubic is the locus of point P such that the circumcevian triangle of P and ABC are orthologic.^[3]

References edit

^ Kimberling, C (1998). "Triangle Centers and Central Triangles". Congress Numerantium. 129: 201.
^ ^a ^b ^c Weisstein, Eric W. ""Circumcevian Triangle"". From MathWorld--A Wolfram Web Resource. MathWorld. Retrieved 24 December 2021.
^ Bernard Gilbert. "K003 McCay Cubic". Catalogue of Triangle Cubics. Bernard Gilbert. Retrieved 24 December 2021.

[1] Kimberling, C (1998). "Triangle Centers and Central Triangles". Congress Numerantium. 129: 201.

[Wolfram-2] Weisstein, Eric W. ""Circumcevian Triangle"". From MathWorld--A Wolfram Web Resource. MathWorld. Retrieved 24 December 2021.

[K003-3] Bernard Gilbert. "K003 McCay Cubic". Catalogue of Triangle Cubics. Bernard Gilbert. Retrieved 24 December 2021.

[1]

[2]

[3]

Summary

Definition edit

Coordinates edit

Some properties edit

See also edit

References edit