Medial triangle: Difference between revisions
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[[Image:Medial Triangle.svg|thumb|right|The red triangle is the medial triangle of the black. The endpoints of the red triangle coincide with the midpoints of the black triangle.]] The '''medial triangle''' of a [[triangle]] ''ABC'' is the triangle with vertices at the [[midpoint]]s of the triangle's sides AB, AC and BC. |
[[Image:Medial Triangle.svg|thumb|right|The red triangle is the medial triangle of the black. The endpoints of the red triangle coincide with the midpoints of the black triangle.]] The '''medial triangle''' or '''midpoint triangle''' of a [[triangle]] ''ABC'' is the triangle with [[vertex (geometry)|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 [[Median (geometry)|medians]] of ''ABC''.) |
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==Properties== |
==Properties== |
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Revision as of 16:09, 29 September 2014
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 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 of ABC.)
Properties
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. 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.
Note that the orthocenter 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.
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.[1]: p.139
Coordinates
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
Anticomplementary triangle
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.
References
- ^ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.