Cubic Curves in The Plane of A Triangle
Suppose that ABC is a triangle with sidelengths a = |BC|, b = |CA|, c = |AB|. Relative to ABC, many named cubics pass through well known points. Examples shown below use two kinds of homogeneous coordinates: trilinear and barycentric.
To convert from trilinear to barycentric in a cubic equation, substitute as follows:
x → bcx, y → cay, z → abz;
to convert from barycentric to trilinear, use
x → ax, y → by, z → cz.
Many equations for cubics have the form
f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,z,x,y) = 0.
In the examples below, such equations are written more succinctly in "cyclic sum notation", like this:
= 0.
The cubics listed below can be defined in terms of the isogonal conjugate, denoted by X*, of a point X not on a sideline of ABC. A construction of X* follows. Let LA be the reflection of line XA about the internal angle bisector of angle A, and define LB and LC analogously. Then the three reflected lines concur in X*. In trilinear coordinates, if X = x:y:z, then X* = 1/x:1/y:1/z.
Read more about this topic: Cubic Plane Curve
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