The medial triangle can also be viewed as the image of triangle 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 . It also follows from this that the
perimeter of the medial triangle equals the
semiperimeter of triangle , and that the
area is one quarter of the area of triangle . 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. The
orthocenter of the medial triangle coincides with the
circumcenter of triangle . 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. In particular, this means that the incenter of a triangle must lie in its medial triangle. The incenter of the medial triangle is the
Spieker center of its reference triangle. 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. The medial triangle is the only
inscribed triangle for which none of the other three interior triangles has smaller area. The reference triangle and its medial triangle are
orthologic triangles. ==Coordinates==