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# Medial Triangle

The triangle formed by joining the midpoints of the sides of a triangle . The medial triangle is sometimes also called the auxiliary triangle (Dixon 1991).

The medial triangle is the Cevian triangle of the triangle centroid and the pedal triangle of the circumcenter (Kimberling 1998, p. 155). It is also the cyclocevian triangle of the orthocenter .

The medial triangle is the polar triangle of the Steiner inellipse.

 (1)

or

 (2)

The medial triangle of a triangle is similar to and its side lengths are

 (3) (4) (5)

This follows immediately by inspecting the construction of the medial triangle and noting that the three vertex triangles and medial triangle each have sides of length , , and . Similarly, each of these triangles, including , have area

 (6)

where is the triangle area of .

The incircle of the medial triangle is called the Spieker circle, and its incenter is called the Spieker center. The circumcircle of the medial triangle is the nine-point circle.

Given a reference triangle , let the angle bisectors of and cut the side (or extended side) of the medial triangle at and . Then is perpendicular to the angle bisector of and is perpendicular to the angle bisector of . Similarly, by taking pairs of angle bisectors in turn, perpendiculars can be dropped from and to their respective intersections with the other sides of the medial triangle (Carding 2006; F. M. Jackson, pers. comm., Aug. 5, 2006).

The following table gives the centers of the medial triangle in terms of the centers of the reference triangle for Kimberling centers with .

 center of medial triangle center of reference triangle incenter Spieker center triangle centroid triangle centroid circumcenter nine-point center orthocenter circumcenter nine-point center midpoint of and symmedian point complement of symmedian point Gergonne point mittenpunkt Nagel point incenter mittenpunkt complement of mittenpunkt Spieker center complement of Feuerbach point complement of first Fermat point complement of second Fermat point complement of first isodynamic point complement of second isodynamic point complement of first Napoleon point complement of second Napoleon point complement of de Longchamps point orthocenter Schiffler point complement of Schiffler point Exeter point complement of far-out point complement of homothetic center of orthic and tangential triangles complementary conjugate of Cevapoint of orthocenter and Clawson center complement of Euler infinity point Euler infinity point second power point complementary conjugate of third power point complement of crosspoint of and isogonal conjugate of Bevan point midpoint of and orthocenter of orthic triangle isogonal conjugate of Kosnita point isogonal conjugate of internal similitude center of circumcircle and incircle complementary conjugate of external similitude center of circumcircle and incircle complementary conjugate of isogonal conjugate of complement of isogonal conjugate of complement of isogonal conjugate of -Ceva conjugate of isogonal conjugate of complementary conjugate of orthocenter of the intouch triangle intersection of lines and isogonal conjugate of -Ceva conjugate of Prasolov point isogonal conjugate of symmedian point of the anticomplementary triangle symmedian point isogonal conjugate of inverse-in-incircle of Jerabek antipode isotomic conjugate of incenter crosspoint of incenter and triangle centroid third Brocard point Brocard midpoint isogonal conjugate of isogonal conjugate of reflection of incenter in Feuerbach point -Ceva conjugate of Cevapoint of incenter and symmedian point isogonal conjugate of isotomic conjugate of isogonal conjugate of Cevapoint of incenter and triangle centroid isogonal conjugate of Cevapoint of incenter and Clawson point isogonal conjugate of Cevapoint of triangle centroid and circumcenter -Ceva conjugate of Tarry point Kiepert antipode Steiner point center of Kiepert hyperbola anticomplement of Feuerbach point Feuerbach point

Anticomplementary Triangle, Circum-Medial Triangle, Cleavance Center, Cleaver, Median Triangle, Nine-Point Circle, Spieker Center, Spieker Circle, Steiner Inellipse, Triangle Median

## References

Carding, M. "Culture Shock for Mathematics and Science." Math. Today 42, 129-131, Aug. 2006.Coxeter, H. S. M. and Greitzer, S. L. "The Medial Triangle and Euler Line." §1.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 18-20, 1967.Dixon, R. Mathographics. New York: Dover, p. 56, 1991.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Medial Triangle

## Cite this as:

Weisstein, Eric W. "Medial Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MedialTriangle.html