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
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

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 .

Carding, M. "Culture Shock for Mathematics and Science." Math. Today42, 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.