Cevian Triangle -- from Wolfram MathWorld
- ️Weisstein, Eric W.
Given a point 
and a triangle 
, the Cevian triangle 
is defined as the triangle composed of the endpoints
of the cevians though the Cevian
point 
.
A triangle and its Cevian triangle are therefore perspective
with respect to the Cevian point. If the point 
has trilinear
coordinates 
,
then the Cevian triangle has trilinear vertex matrix
![[0 beta gamma; alpha 0 gamma; alpha beta 0]](https://mathworld.wolfram.com/images/equations/CevianTriangle/NumberedEquation1.svg) |
(1) |
(Kimberling 1998, pp. 55 and 185), and is a central triangle of type 1 (Kimberling 1998, p. 55).
The following table summarizes a number of special Cevian triangles for various special Cevian points 
.
If 
is the Cevian triangle of 
and 
is the anticevian
triangle, then 
and 
are harmonic conjugates with respect to 
and 
.
The side lengths of the Cevian triangle with respect to a Cevian point 
are given by
The area of the Cevian triangle of 
with respect to the center with trilinear coordinates
is given by
 |
(5) |
where 
is the area of triangle 
.
If 
is the Cevian triangle of 
, then the triangle 
obtained by reflecting 
, 
, and 
across the midpoints of their sides is also a Cevian triangle
of 
(Honsberger 1995, p. 141; left figure). Furthermore, if the Cevian
circle crosses the sides of 
in three points 
, 
, and 
, then 
is also a Cevian triangle of 
(Honsberger 1995, pp. 141-142; right figure).
See also
Anticevian Triangle, Cevian, Cevian Circle, Cevian Point
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References
Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 141-143, 1995.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.
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Cite this as:
Weisstein, Eric W. "Cevian Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CevianTriangle.html