Barycentric Coordinates

Barycentric coordinates are the natural coordinate system for the triangle. They express points as "weighted averages" of the vertices $A, B, C$ .

Definition

Definition (Barycentric Coordinates)

For a reference triangle

A BC

, any point

P

can be written uniquely as:

P = \frac{u A + v B + wC}{u + v + w}

The coordinates are usually denoted

(u : v : w)

, which are homogeneous (scaling all by

k

represents the same point).

If we require

u + v + w = 1

, then

u, v, w

are the areal coordinates:

u = \frac{Area ( PBC )}{Area ( A BC )}, v = \frac{Area ( PC A )}{Area ( A BC )}, w = \frac{Area ( P A B )}{Area ( A BC )}

A line in barycentric coordinates has the form:

p x + q y + rz = 0

Exercise (Problem 1)

Find the barycentric coordinates of the excenter

I_{a}

Exercise (Problem 2)

Prove that the midpoints of the sides are

M_{a} = (0 : 1 : 1)

M_{b} = (1 : 0 : 1)

, etc.

Exercise (Problem 3)

Determine the condition for three points

P_{1} (u_{1} : v_{1} : w_{1})

P_{2}

P_{3}

to be collinear (Determinant).