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Medial triangle

## Summary

The medial triangle or midpoint triangle of a triangle ABC is the triangle with vertices at the midpoints of the triangle's sides AB, AC and BC. It is the n=3 case of the midpoint polygon of a polygon with n sides. The medial triangle is not the same thing as the median triangle, which is the triangle whose sides have the same lengths as the medians of ABC.

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

Each side of the medial triangle is called a midsegment (or midline). In general, a midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle. It is parallel to the third side and has a length equal to half the length of the third side.

## Properties

M: circumcenter of ${\displaystyle \triangle ABC}$ , orthocenter of ${\displaystyle \triangle DEF}$
N: incenter of ${\displaystyle \triangle ABC}$ , Nagel point of ${\displaystyle \triangle DEF}$
S: centroid of ${\displaystyle \triangle ABC}$  and${\displaystyle \triangle DEF}$

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. Thus, the sides of the medial triangle are half and parallel to the corresponding sides of triangle ABC. Hence, the medial triangle is inversely similar and shares the same centroid and medians with triangle ABC. It also follows from this that the perimeter of the medial triangle equals the semiperimeter of triangle ABC, and that the area is one quarter of the area of triangle ABC. Furthermore, the four triangles that the original triangle is subdivided into by the medial triangle are all mutually congruent by SSS, so their areas are equal and thus the area of each is 1/4 the area of the original triangle.[1]: p.177

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. The nine-point circle circumscribes the medial triangle, and so the nine-point center is the circumcenter of the medial triangle.

The Nagel point of the medial triangle is the incenter of its reference triangle.[2]: p.161, Thm.337

A reference triangle's medial triangle is congruent to the triangle whose vertices are the midpoints between the reference triangle's orthocenter and its vertices.[2]: p.103, #206, p.108, #1

The incenter of a triangle lies in its medial triangle.[3]: p.233, Lemma 1

A point in the interior of a triangle is the center of an inellipse of the triangle if and only if the point lies in the interior of the medial triangle.[4]: p.139

The medial triangle is the only inscribed triangle for which none of the other three interior triangles has smaller area.[5]: p. 137

The reference triangle and its medial triangle are orthologic triangles.

## Coordinates

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

## Anticomplementary triangle

If XYZ is the medial triangle of ABC, then ABC is the anticomplementary triangle or antimedial 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.

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.