Theorems on Quadriplanar Coordinates

Extract

Given a tetrahedron of reference ABCD, let λ, μ, ν, π be the cosines of the angles made by a line with the perpendiculars to the faces BCD, ACD, ABD, ABC respectively, these perpendiculars being drawn all inwards or all outwards; where

Aλ₁ + Bμ₁ + C½₁ + Dπ₁ = 0

Aλ₂ + Bμ₂ + C½₂ + Dπ₂ = 0

A, B, C, D being the areas of the faces BCD, .. If P₁ (α₁, ²₁, γ₁, δ₁) and P₂ (α₂, β₂, γ₂, δ₂) are any two points, the lines PP₁ and PP₂ can be written

(α - α₁)/λ₁ = (β - β₁)/μ₁ = (γ - γ₁)/ν₁ = (δ - δ₁)/π₁ = θ₁,

(α - α₂)/λ₂ = (β - β₂)/μ₂ = (γ - γ₂)/ν₂ = (δ - δ₂)/π₂ = θ₂,

to find the cosine of the angle between PP₁ and PP₂.