Any point in a triangle can be named by how much it leans toward each corner — three weights that always sum to one. At a corner the weight is (1,0,0); dead center it’s a three-way tie. Step outside and a weight goes negative. It’s how graphics shade a triangle and how you interpolate between three references. Slide the point and read its pull.
Barycentric coordinates write a point P as a weighted average of the triangle’s vertices, P = αA + βB + γC with α+β+γ = 1. The weights are ratios of sub-triangle areas, so they are exactly 1 at the matching vertex and P is INSIDE the triangle if and only if all three are non-negative; a negative weight means P is beyond the opposite edge. A fail-loud self-check throws unless the weights always sum to 1, an interior point has all three ≥ 0, and an exterior point has a negative one.
Degenerate (collinear) triangles have no valid barycentric frame (the area denominator vanishes). The area-ratio formula, the sum-to-one, and the inside/outside test are exact.