Medial Triangle

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

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

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

Its trilinear vertex matrix is

or

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

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 a/2, b/2, and c/2. Similarly, each of these triangles, including DeltaM_AM_BM_C, have area

where Delta is the triangle area of DeltaABC.

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 DeltaABC, let the angle bisectors of A and B cut the side (or extended side) of the medial triangle DeltaM_AM_B at I_A and I_B. Then CI_A is perpendicular to the angle bisector of A and CI_B is perpendicular to the angle bisector of B. Similarly, by taking pairs of angle bisectors in turn, perpendiculars can be dropped from A and B 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 X_n with n<=100.

See also

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References

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Cite this as:

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

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