The medial triangle of a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 ABC is the triangle with vertices at the midpoint

Midpoint

The midpoint is the middle point of a line segment. It is equidistant from both endpoints.-Formulas:...

s of the triangle's sides AB, AC and BC. (The medial triangle is different from the median triangle.)

The medial triangle can also be viewed as the image of triangle ABC transformed by a homothety centered at the centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

 with ratio -1/2. Hence, the medial triangle is inversely similar

Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

 and shares the same centroid and medians

Median (geometry)

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side...

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

Perimeter

A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 of the medial triangle equals the semiperimeter

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 name...

 of triangle ABC, and that the area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 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

Line (geometry)

The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

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

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 three sides of the triangle...

 of the circumcenter.

If XYZ is the medial triangle of ABC, then ABC is the anticomplementary 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.

Let a = |BC|, b = |CA|, c = |AB| be the sidelengths of triangle ABC. Trilinear coordinates

Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous 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

Trilinear coordinates

Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous 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.

A point in the interior of a triangle is the center of an inellipse

Circumconic and inconic

In triangle geometry, a circumconic is a conic section that passes through three given points, and an inconic is a conic section inscribed in the triangle whose vertices lie at three given points....

of the triangle if and only if the point lies in the interior of the medial triangle.