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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} .
In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle ABC, let the lines AO, BO, CO be drawn from the vertices to a common point O (not on one of the sides of ABC), to meet opposite sides at D, E, F respectively. (The segments AD, BE, CF are known as cevians.) Then, using signed lengths of segments,
The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. [ 64 ] As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the ...
Möbius's original formulation of homogeneous coordinates specified the position of a point as the center of mass (or barycenter) of a system of three point masses placed at the vertices of a fixed triangle. Points within the triangle are represented by positive masses and points outside the triangle are represented by allowing negative masses.
The term triangular coordinates may refer to any of at least three related systems of coordinates in the Euclidean plane: . a special case of barycentric coordinates for a triangle, in which case it is known as a ternary plot or areal coordinates, among other names
If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (x B, y B) and C = (x C, y C), then the area can be computed as 1 ⁄ 2 times the absolute value of the determinant
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]