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 midpoints of the triangle's sides . It is the case of the midpoint polygon 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 ...

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 The Medes (Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, the ...

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 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

with triangle . It also follows from this that the perimeter 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 ...

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 op ...

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 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 of the medial triangle coincides with the circumcenter of triangle . This fact provides a tool for proving collinearity 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 ...

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 to the triangle whose vertices are the midpoints between the reference triangle's orthocenter 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 side ...

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 i ...

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 t ...

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