Since Descartes, it has been customary to specify a point in the Euclidean plane by its two coordinates in an arbitrary coordinate system. But when studying the properties of a triangle analytically, a geometer guided by sound aesthetic principles will balk at breaking its intrinsic symmetry by choosing one vertex as the origin or one side as a basis vector. Any result should be expressed in a form invariant under every permutation of the triangle’s vertices. Three coordinates are therefore needed, linked by a relation (since the problem is planar and hence two-dimensional). This is the principle behind trilinear coordinates and barycentric coordinates, which we shall use here to identify remarkable points and curves associated with a triangle.
Triangle coordinates
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For any point P in the plane determined by the non-collinear points A, B and C, the three vectors and are linearly dependent and therefore satisfy a relation of the form
The point P is then called the barycenter of the points (A, B, C), assigned the respective coefficients (α, β, γ). To indicate that the barycenter is unchanged when (α, β, γ) is replaced by proportional values (μα, μβ, μγ), we may write either P (μα : μβ : μγ) or P (α : β : γ).
This explains Möbius’s use of the term homogeneous coordinates in 1827. A homogeneous function of degree p is a function f of n variables such that
f (λ x1, λ x2… λ *xn ) = λ p f (x*1, x2… xn ) for every real number λ, and barycentric coordinates are homogeneous of degree 0 because (λα : λβ : λγ) = λ0(α : β : γ).
