The Barycentric Coordinate System is a method used in geometry to express the position of a point within a triangle or a simplex. In this system, any point inside the triangle can be represented as a weighted average of the triangle's vertices, with the weights corresponding to the area of the sub-triangles formed with the point and the vertices. In a triangle defined by vertices A, B, and C, a point P can be expressed using barycentric coordinates (α, β, γ), where α, β, and γ are the weights assigned to vertices A, B, and C, respectively. The sum of these weights equals one (α + β + γ = 1), allowing for a clear representation of the point's relative position within the triangle.