Exact Trilinear Coordinates

The Trilinear Coordinates $\alpha:\beta:\gamma$ of a point $P$ relative to a Triangle are Proportional to the directed distances $a':b':c'$ from $P$ to the side lines (i.e, $a'=k\alpha$, $b'=k\beta$, $c'=k\gamma$). Letting $k$ be the constant of proportionality,

$$k\equiv {2\Delta\over a\alpha+b\beta+c\gamma},$$

where $\Delta$ is the Area of $\Delta ABC$ and $a$, $b$, and $c$ are the lengths of its sides. When the trilinears are chosen so that $k=1$, the coordinates are known as exact trilinear coordinates.

See also Trilinear Coordinates