The medial triangle or midpoint triangle of a triangle
''ABC'' is the triangle with vertices
at the midpoint
s 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
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.
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
, so their areas are equal and thus the area of each is 1/4 the area of the original triangle.
[Posamentier, Alfred S., and Lehmann, Ingmar. ''The Secrets of Triangles'', Prometheus Books, 2012.]
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.
[Altshiller-Court, Nathan. ''College Geometry''. Dover Publications, 2007.]
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 incenter of a triangle lies in its medial triangle.] [Franzsen, William N.. "The distance from the incenter to the Euler line", ''Forum Geometricorum'' 11 (2011): 231–236.]
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.
[Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.]
The medial triangle is the only inscribed triangle for which none of the other three interior triangles has smaller area. [ Torrejon, Ricardo M. "On an Erdos inscribed triangle inequality", ''Forum Geometricorum'' 5, 2005, 137–141. http://forumgeom.fau.edu/FG2005volume5/FG200519index.html]
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
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.
*Middle hedgehog, an analogous concept for more general convex sets
Category:Objects defined for a triangle
de:Mittelparallele#Mittelparallelen eines Dreiecks