- Steve Phelps
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 of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio x:y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y:z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z:x and vertices C and A. (https://en.wikipedia.org/wiki/Trilinear_coordinates)
This interactive figure shows the lattice of the trilinear coordinate system for triangle ABC. You can clearly see the interior and exterior angle bisectors. You can also clearly see other lines and points of concurrency. This lattice system is based on the incenter.
Conics play an important role in the geometry of the triangle. Any five points determine a conic. Notice that a conic that passes through any five of the trilinear lattice points tend to pass through others.
In the context of a triangle, barycentric coordinates are also known as area coordinates or areal coordinates, because the coordinates of P with respect to triangle ABC are equivalent to the (signed) ratios of the areas of PBC, PCA and PAB to the area of the reference triangle ABC. Areal and trilinear coordinates are used for similar purposes in geometry. (https://en.wikipedia.org/wiki/Barycentric_coordinate_system) This lattice system is based on the centroid.