Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, a circumcevian triangle is a special

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

associated with a reference triangle and a point in the plane of the triangle. It is also associated with the

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

of the reference triangle.

Definition

Let be a point in the plane of the reference triangle . Let the lines intersect the

of at . The triangle is called the circumcevian triangle of with reference to .

Coordinates

Let be the side lengths of triangle and let the

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...

of be . Then the trilinear coordinates of the vertices of the circumcevian triangle of are as follows:

\begin
  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

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 In plane geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle , and a point that is not one of the vertices . Drop perpendiculars from to the three sides of the tr ...

of P. *The

McCay cubic In Euclidean geometry, the McCay cubic (also called M'Cay cubic or Griffiths cubic) is a cubic plane curve in the plane of a reference triangle and associated with it. It is the third cubic curve in Bernard Gilbert's Catalogue of Triangle Cubics ...

is the locus of point P such that the circumcevian triangle of P and ABC are orthologic.

References

{{reflist Triangle geometry

Definition

Coordinates

Some properties

See also

References