Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, the medial triangle or midpoint triangle of a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

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''-dimen ...

s of the triangle's sides . It is the case of the

midpoint polygon In geometry, the midpoint polygon of a polygon is the polygon whose vertices are the midpoints of the edges of . It is sometimes called the Kasner polygon after Edward Kasner, who termed it the ''inscribed polygon'' "for brevity". Examples T ...

of a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

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 ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by th ...

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 surface of the figure. The same definition extends to any ...

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 a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pr ...

of the medial triangle equals the

semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate na ...

of triangle , and that the

area Area is the quantity that expresses the extent of a region on the plane or on a curved 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 ope ...

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 mod ...

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 In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...

of the medial triangle coincides with the

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

of triangle . This fact provides a tool for proving

collinearity In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...

of the circumcenter, centroid and orthocenter. The medial triangle is the

pedal triangle In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the th ...

of the circumcenter. The

nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of ea ...

circumscribes the medial triangle, and so the nine-point center is the circumcenter of the medial triangle. The

Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concu ...

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

to the triangle whose vertices are the midpoints between the reference triangle's

and its vertices. The

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 In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, t ...

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 In geometry, two triangles are said to be orthologic if the perpendiculars from the vertices of one of them to the corresponding sides of the other are concurrent (i.e., they intersect at a single point). This is a symmetric property; that is, ...

Coordinates

Let

a = , BC, , b = , CA, , c = , AB,

be the sidelengths of triangle .

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 are given by :

\begin
X &= 0 : \frac : \frac \\
Y &= \frac : 0 : \frac \\
Z &= \frac : \frac : 0
\end

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 .

for the vertices of the anticomplementary triangle, , are given by :

\begin
X' &= \frac : \frac : \frac \\
Y' &= \frac : \frac : \frac \\
Z' &= \frac : \frac : \frac
\end

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 .

References

External links

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

Properties

Coordinates

Anticomplementary triangle

See also

References

External links