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More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors ,, and if the point P has trilinear coordinates x : y : z, then the Cartesian coordinates of are the weighted average of the Cartesian coordinates of these vertices using the barycentric ...
The Nagel triangle or extouch triangle of is denoted by the vertices , , and that are the three points where the excircles touch the reference and where is opposite of , etc. This T A T B T C {\displaystyle \triangle T_{A}T_{B}T_{C}} is also known as the extouch triangle of A B C {\displaystyle \triangle ABC} .
The red triangle is the medial triangle of the black. The endpoints of the red triangle coincide with the midpoints of the black triangle. In Euclidean geometry, 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, BC.
It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos −1 ( 1 / 5 ), or approximately 78.46°. The 5-simplex is a solution to the problem: Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.
The area of the extouch triangle, K T, is given by: = where K and r are the area and radius of the incircle, s is the semiperimeter of the original triangle, and a, b, c are the side lengths of the original triangle. This is the same area as that of the intouch triangle. [2]
A lattice triangle is any triangle drawn within a 2D lattice such that all vertices lie on lattice points. By Pick's theorem a lattice triangle has a rational area that either is an integer or a half-integer (has a denominator of 2). If the lattice triangle has integer sides then it is Heronian with integer area. [20]