Circum-Medial Triangle


The circum-medial triangle DeltaA^'B^'C^' is the circumcevian triangle of a reference triangle DeltaABC with respect to the triangle centroid G of DeltaABC (Kimberling 1998, pp. 162-163).

It has trilinear vertex matrix

 [-abc c(b^2+c^2) b(b^2+c^2); c(c^2+a^2) -abc a(c^2+a^2); b(a^2+b^2) a(a^2+b^2) -abc].

Its area is


where Delta is the area of DeltaABC.

The following table gives the centers of the circum-medial triangle in terms of the centers of the reference triangle that correspond to Kimberling centers X_n.

X_ncenter of circum-medial triangleX_ncenter of reference triangle
X_6symmedian pointX_(183)trilinear quotient X_(75)/X_(98)
X_(74)X_(74)X_(2696)sr(X_(74), X_(111))
X_(110)focus of Kiepert parabolaX_(2770)sr(X_(110), X_(111))
X_(511)isogonal conjugate of X_(98)X_(2782)isogonal conjugate of X_(2698)
X_(512)isogonal conjugate of X_(99)X_(804)odd (-2,2) infinity point
X_(1379)first Brocard-axis intercept of circumcircleX_(99)Steiner point
X_(1380)second Brocard-axis intercept of circumcircleX_(98)Tarry point

See also

Circumcevian Triangle, Medial Triangle

Explore with Wolfram|Alpha


Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Circum-Medial Triangle

Cite this as:

Weisstein, Eric W. "Circum-Medial Triangle." From MathWorld--A Wolfram Web Resource.

Subject classifications