Medial Triangle
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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, the medial triangle or midpoint triangle of a
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
is the triangle with vertices at the
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...
s of the triangle's sides . It is the case of the midpoint polygon of a
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
with 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 . 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

The medial triangle can also be viewed as the image of triangle transformed by a
homothety In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point called its ''center'' and a nonzero number called its ''ratio'', which sends point to a point by the rule, : \o ...
centered at the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...
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 A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...
of the medial triangle equals the semiperimeter of triangle , and that the
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
by SSS, 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 ''The Secrets of Triangles: A Mathematical Journey'' is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and , and published in 2012 by Prometheus Books. Topics The book consists of ten chapters, ...
'', Prometheus Books, 2012.
The
orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...
of the medial triangle coincides with the
circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...
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 In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...
of its reference triangle.Altshiller-Court, Nathan. ''College Geometry''. Dover Publications, 2007. A reference triangle's medial triangle is
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
to the triangle whose vertices are the midpoints between the reference triangle's
orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...
and its vertices. The
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...
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.
/ref> 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 The reference triangle and its medial triangle are orthologic triangles.


Coordinates

Let a = , BC, , b = , CA, , c = , AB, be the sidelengths of triangle \triangle ABC.
Trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...
for the vertices of the medial triangle \triangle EFD are given by :\begin E &= \, 0 &&: \frac &&: \frac, \\ muF &= \frac &&: \,0 &&: \frac, \\ muD &= \frac &&: \frac &&: \, 0. \end


Anticomplementary triangle

If \triangle EFD is the medial triangle of \triangle ABC, then \triangle ABC is the anticomplementary triangle or antimedial triangle of \triangle EFD. The anticomplementary triangle of \triangle ABC is formed by three lines parallel to the sides of the parallel to AB through C, the parallel to AC through B, and the parallel to BC through A.
Trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...
for the vertices of the triangle \triangle E'F'D' anticomplementary to \triangle ABC are given by :\begin E' &= -\frac &&: \phantom\frac &&: \phantom\frac, \\ muF' &= \phantom\frac &&: -\frac &&: \phantom\frac, \\ muD' &= \phantom\frac &&: \phantom\frac &&: -\frac. \end 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.


See also

* Middle hedgehog, an analogous concept for more general convex sets


References


External links

* * {{DEFAULTSORT:Medial Triangle Elementary geometry Objects defined for a triangle de:Mittelparallele#Mittelparallelen eines Dreiecks