Medial triangle
In Euclidean geometry, the medial triangle or midpoint triangle of a triangle Template:Math is the triangle with vertices at the midpoints of the triangle's sides Template:Mvar, Template:Mvar, and Template:Mvar. It is the Template:Math case of the midpoint polygon of a polygon with Template:Mvar 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 Template:Math.
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
Template:Mvar: incenter of Template:Math, Nagel point of Template:Math
Template:Mvar: centroid of Template:Math and Template:Math
The medial triangle can also be viewed as the image of triangle Template:Math 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 Template:Math. Hence, the medial triangle is inversely similar and shares the same centroid and medians with Template:Math. It also follows from this that the perimeter of the medial triangle equals the semiperimeter of Template:Math, and that the area is one quarter of the area of Template:Math. 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.[1]Template:Rp
The orthocenter of the medial triangle coincides with the circumcenter of Template:Math. 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.[2]Template:Rp In particular, this means that the incenter of a triangle must lie in its medial triangle.[3]Template:Rp
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.[2]Template:Rp
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]Template:Rp
The medial triangle is the only inscribed triangle for which none of the other three interior triangles has smaller area.[5]Template:Rp
The reference triangle and its medial triangle are orthologic triangles.
Coordinates
Let Template:Math, Template:Math, Template:Math be the sidelengths of triangle Template:Math. Trilinear coordinates for the vertices of the medial triangle Template:Math are given by:
Anticomplementary triangle
If Template:Math is the medial triangle of Template:Math, then Template:Math is the anticomplementary triangle or antimedial triangle of Template:Math. The anticomplementary triangle of Template:Math is formed by three lines parallel to the sides of Template:Math: the parallel to Template:Mvar through Template:Mvar, the parallel to Template:Mvar through Template:Mvar, and the parallel to Template:Mvar through Template:Mvar.
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 Template:Mvar, Template:Mvar, and Template:Mvar of the reference triangle. The vertices of the medial triangle are the complements of Template:Mvar, Template:Mvar, and Template:Mvar.
See also
- Middle hedgehog, an analogous concept for more general convex sets
References
External links
de:Mittelparallele#Mittelparallelen eines Dreiecks
- ↑ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
- ↑ 2.0 2.1 Altshiller-Court, Nathan. College Geometry. Dover Publications, 2007.
- ↑ Template:Cite journal
- ↑ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.
- ↑ Template:Cite journal