Triangle center

Remarks

Trilinear coordinates $\alpha : \beta : \gamma$ describe the point whose signed distances to the lines $BC$, $CA$, $AB$ are in the ratio $\alpha : \beta : \gamma$. Homogeneity makes the point move with the triangle under scaling, and bisymmetry makes it independent of the labelling of the vertices.

Example: $f(a, b, c) = 1$ is homogeneous of degree $0$ and bisymmetric. Its center $1 : 1 : 1$ is equidistant from the three sides, so it is the incentre. Similarly $f(a, b, c) = 1/a$ gives the centroid.

Non-example: $f(a, b, c) = c/b$ is homogeneous but not bisymmetric, since $f(a, c, b) = b/c$. The point $c/b : a/c : b/a$ is the first Brocard point, which changes into the second Brocard point when two vertices are swapped, so it is not a triangle center.