Medial triangle
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Medial_Triangle.svg/220px-Medial_Triangle.svg.png)
In Euclidean geometry, 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, 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.
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
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Mittendreieck.svg/350px-Mittendreieck.svg.png)
N: incenter of △ABC, Nagel point of △DEF
S: centroid of △ABC and △DEF
The medial triangle can also be viewed as the image of triangle △ABC transformed by a
The
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
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 be the sidelengths of triangle Trilinear coordinates for the vertices of the medial triangle are given by
Anticomplementary triangle
If is the medial triangle of then is the anticomplementary triangle or antimedial triangle of The anticomplementary triangle of is formed by three lines parallel to the sides of : the parallel to through the parallel to through and the parallel to through
Trilinear coordinates for the vertices of the triangle anticomplementary to are given by
The name "anticomplementary triangle" corresponds to the fact that its vertices are the anticomplements of the vertices of the reference triangle. The vertices of the medial triangle are the complements of
See also
- Middle hedgehog, an analogous concept for more general convex sets
References
- ^ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
- ^ a b Altshiller-Court, Nathan. College Geometry. Dover Publications, 2007.
- ^ Franzsen, William N.. "The distance from the incenter to the Euler line", Forum Geometricorum 11 (2011): 231–236.
- ^ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.
- ^ Torrejon, Ricardo M. "On an Erdos inscribed triangle inequality", Forum Geometricorum 5, 2005, 137–141. http://forumgeom.fau.edu/FG2005volume5/FG200519index.html