104.09 A harmonic range for the triangle

TLDR: In this paper, the Feuerbach line is shown to be related to the points on the following relative coordinates: F (λ (μ − ν), μ (ν − λ), ν (λ − μ)), I (1 − μν, 1− νλ, 1 − ǫμ), N (1 + βγ, 1+ γα, 1 + αβ).Then, to see how these points are related, we need only work with one coordinate (, say).

Abstract: (x + y + z = 1) α = cot A β = cot B γ = cot C (βγ + γα + αβ = 1) λ = tan 2A μ = tan 2B ν = tan 2C (μν + νλ + λμ = 1, Σ = λ + μ + ν, Π = λμν) 2α = 1 λ − λ 2β = 1 μ − μ 2γ = 1 ν − ν [] () H [βγ + γα + αβ] G (1, 1, 1) Now further relative coordinates are F (λ (μ − ν) , μ (ν − λ) , ν (λ − μ)) , I (1 − μν, 1 − νλ, 1 − λμ) , N (1 + βγ, 1 + γα, 1 + αβ) . Then, to see how these points on the Feuerbach line are related, we need only work with one coordinate ( , say). x Thus for : N