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X₁₃₅₄–X₁₃₉₂¶

Brisse Transforms, Rigby/Griffiths Points & Complementary Conjugates

Polynomial centers on the incircle generated by the Brisse transform (X₁₃₅₄–X₁₃₆₇), Rigby and Griffiths points, and complementary and anticomplementary conjugates.

X₁₃₅₄–X₁₃₉₂ (Brisse Transforms, Rigby/Griffiths Points & Complementary Conjugates) — 15 representative centers (from 39 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — Brisse Transforms 1354-1367

Suppose that P is a point on the circumcircle Γ of triangle ABC. Let U and V be the lines through P tangent to the incircle. Line U meets Γ in a point U' other than P, and line V meets Γ in a point V' other than P. The line U'V' is tangent to the incircle. The touchpoint, denoted by T(P), is the Brisse transform of P. Suppose P is given by barycentrics \(u : v : w\). Barycentrics for T(P) are found in

Edward Brisse, Perspective Poristic Triangles: \(a^{4}\)/[(b + c - a)\(u^{2}\)] : \(b^{4}\)/[(c + a - b)\(v^{2}\)] : \(c^{4}\)/[(a + b - c)\(w^{2}\)].

If X is given by trilinears \(x : y : z\), then T(X) has trilinears a/[(b + c - a)\(x^{2}\)] : b/[(c + a - b)\(y^{2}\)] : c/[(a + b - c)\(z^{2}\)].

Examples: X₁₁ = Feuerbach point = T(X₁₀₉)

X₁₃₁₇ = incircle-antipode of X₁₁ = T(X₁₀₆)

Still open is the question posed in Hyacinthos #6832: to list all polynomial centers on the incircle having low degree and to prove that there are no others. Here, “degree” of X =\(p(a,b,c) : p(b,c,a) : p(c,a,b)\) [barycentrics] refers to the degree of homogeneity of p(a,b,c), and “low” means less than 6. (The Feuerbach point, X₁₁, has degree 3.)

In Hyacinthos #6835, Paul Yiu gives two methods for constructing polynomial centers on the incircle:

If X is a polynomial center on the incircle and W is any other polynomial center, then the line XW meets the incircle in another point that is a polynomial center.
If W is on the line at infinity, then the barycentric square \(W^{2}\) is on the Steiner inscribed ellipse, and the barycentric product X₇*\(W^{2}\) is on the incircle.