Medial triangle

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The red triangle is the medial triangle of the black. The endpoints of the red triangle coincide with the midpoints of the black triangle.

The medial triangle of a triangle ABC is the triangle with vertices as the midpoints of triangle ABC.

The medial triangle can also be viewed as the image of triangle ABC transformed by a homothety centered at the centroid with ratio -1/2. Hence, the medial triangle is inversely similar and shares the same centroid and medians with triangle ABC.

Note that the orthocenter of the medial triangle coincides with the circumcenter of triangle ABC. This fact provides a tool for proving collinearity of the circumcenter, centroid and orthocenter. The medial triangle is the pedal triangle of the circumcenter.

If XYZ is the medial triangle of ABC, then ABC is the anticomplementary triangle of XYZ. The anticomplementary triangle of ABC is formed by three lines parallel to the sides of ABC: the parallel to AB through C, the parallel to AC through B, and the parallel to BC through A.

Let a = |BC|, b = |CA|, c = |AB| be the sidelengths of triangle ABC. Trilinear coordinates for the vertices of the medial triangle are given by

X = 0 : 1/b : 1/c
Y = 1/a : 0 : 1/c
Z = 1/a : 1/b : 0

Trilinear coordinates for the vertices of the anticomplementary triangle, X'Y'Z', are given by

X' = -1/a : 1/b : 1/c
Y' = 1/a : -1/b : 1/c
Z' = 1/a : 1/b : -1/c

The name "anticomplementary triangle" corresponds to the fact that its vertices are the anticomplements of the vertices A, B, C of the reference triangle. The vertices of the medial triangle are the complements of A, B, C. Finally, the medial triangle is different from the median triangle.