In geometry, the trilinear coordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines...

signed distances from p to these three lines. These are called the trilinear coordinates of p with respect to the triangle whose vertices are the pairwise...

and trilinear coordinates, the various other equations given in Trilinear coordinates#Formulas can be rewritten in terms of barycentric coordinates. The...

of the triangle, the trilinear coordinates for the incenter are   1 : 1 : 1. {\displaystyle \ 1:1:1.} The barycentric coordinates for a point in a triangle...

on a rectilinear 3D grid Trilinear map, a type of mathematical function between vector spaces Trilinear coordinates Trilinear polarity, in geometry This...

line at infinity, given in trilinear coordinates by a x + b y + c z = 0 {\displaystyle ax+by+cz=0} and in barycentric coordinates by x + y + z = 0. {\displaystyle...

bisector line. The trilinear coordinates for a point in the triangle give the ratio of distances to the triangle sides. Trilinear coordinates for the incenter...

|AN|^{2}+|BL|^{2}+|CM|^{2}=|NB|^{2}+|LC|^{2}+|MA|^{2}.} If P has trilinear coordinates p : q : r, then the vertices L, M, N of the pedal triangle of P...

point of concurrency is the Steiner point of triangle ABC. The trilinear coordinates of the Steiner point are given below. b c / ( b 2 − c 2 ) : c a...

are tetrahedra (3D simplices), then barycentric coordinates provide a straightforward procedure. Trilinear interpolation is frequently used in numerical...

as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center...

as a pencil of coaxal circles. If the circles are represented in trilinear coordinates in the usual way, then their radical center is conveniently given...

and AB (c) are da, db and dc, respectively, the Miquel point, in trilinear coordinates (x : y : z), is given by: x = a ( − a 2 d a d a ′ + b 2 d a d b...

analysis of triangles. Trilinear coordinates are used in the context of triangles. There are ways of describing curves without coordinates, using intrinsic...

Centers the circle E is the called the Apollonius circle of △ABC. The trilinear coordinates of the Apollonius point are a ( b + c ) 2 b + c − a : b ( c + a...

are the side-lengths of a reference triangle then the point whose trilinear coordinates are f ( a , b , c ) : f ( b , c , a ) : f ( c , a , b ) {\displaystyle...

homogeneous coordinates (1 : i : 0) and (1 : −i : 0). Let A. B. C be the measures of the vertex angles of the reference triangle ABC. Then the trilinear coordinates...

The trilinear coordinates of these points cannot be obtained by plugging in the values 0 and 1 for r in the expressions for the trilinear coordinates for...

reference triangle △ABC. Let the trilinear coordinates of the point P be p : q : r. Then the trilinear equation of the trilinear polar of P is x p + y q + z...

{a+b+c}{2}}} is the semi-perimeter of the reference triangle △ABC. The trilinear coordinates of the Nagel point are as csc 2 ⁡ ( A 2 ) : csc 2 ⁡ ( B 2 ) : csc...

therefore the ratio of weights at A and C being α : γ. In trilinear coordinates, a point with coordinates x :y :z has perpendicular distances to side BC (across...

concurrent. The point of concurrence is the Exeter point of △ABC. The trilinear coordinates of the Exeter point are a ( b 4 + c 4 − a 4 ) : b ( c 4 + a 4 −...

concurrence is called the second Morley center of triangle △ABC. The trilinear coordinates of the first Morley center of triangle △ABC are cos ⁡ A 3 + 2 cos...

ternary plot correspond (up to a constant) to its trilinear coordinates or barycentric coordinates. There are three equivalent methods that can be used...

three constructed equilateral triangles are concurrent at X(13). Trilinear coordinates for the first isogonic center, X(13): csc ⁡ ( A + π 3 ) : csc ⁡...

of Triangle Cubics. The equation of a general triangle conic in trilinear coordinates x : y : z has the form r x 2 + s y 2 + t z 2 + 2 u y z + 2 v z x...

is known as a ternary plot or areal coordinates, among other names Trilinear coordinates, in which the coordinates of a point in a triangle are its relative...

similar to the triangles AOI, BOI, COI respectively.: Propos. 4  The trilinear coordinates for the Feuerbach point are 1 − cos ⁡ ( B − C ) : 1 − cos ⁡ ( C...

circle, and the circumcircle of its tangential triangle are coaxal. Trilinear coordinates for the center of the Kiepert hyperbola are ( b 2 − c 2 ) 2 a :...

and having endpoints on these sidelines have equal lengths. The trilinear coordinates of the equal parallelians point of triangle △ABC are b c ( c a +...