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.
Its trilinear vertex matrix is
(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 .