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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 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
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 ...
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 ...
In this case, the coordinates of the vertices ′ = and ′ = represent the vectors from vertex A' to these vertices. Observe that this trivial translation is possible for all triangles and the circumcenter U ′ = ( U x ′ , U y ′ ) {\displaystyle U'=(U'_{x},U'_{y})} of the triangle A'B'C' follow as
In an obtuse triangle (one with an obtuse angle), the foot of the altitude to the obtuse-angled vertex falls in the interior of the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite extended side, exterior to the triangle. This is illustrated in the adjacent diagram: in this obtuse triangle, an ...
Fig 1. Construction of the first isogonic center, X(13). When no angle of the triangle exceeds 120°, this point is the Fermat point. In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible [1] or ...
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