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X₂₁₂₀–X₂₁₄₇¶

Eigencenters & Eigentransforms

Eigentransforms ET(X) — eigencenters of the cevian triangles of classical centers — paired with their isogonal conjugates on the self-isogonal cubic Z(X) (X₂₁₂₀–X₂₁₄₃), followed by X₂-eigentransforms (X₂₁₄₄–X₂₁₄₇).

X₂₁₂₀–X₂₁₄₇ (Eigencenters & Eigentransforms) — 16 representative centers (from 28 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — Eigencenters I, 2120 -2143 (Eigencenters and eigentransforms)

The eigencenter of a triangle is defined in the Glossary and in TCCT, p. 192. In case the triangle is the cevian triangle of a point U =\(u : v : w\), the eigencenter is given by

\(f(u,v,w) : f(v,w,u) : f(w,u,v)\), where \(f(u,v,w) = vw(u^{2}v^{2} + u^{2}w^{2} - v^{2}w^{2})\).

This point is now named the eigentransform of U, denoted by ET(U). Pairs (i,j) such that X_j = ET(X_i) include

(1,1), (2,3), (3,1075), (4,155), (6,194), (7,218), (13,62), (14,61), (63,1712), (88,1), (92,47), (94,49), (99,39), (100,1), (110,5), (162,1), (174,266), (190,1), (648,185), (651,1), (653,1), (655,1), (658,1), (660,1), (662,1), (664,2082), (673,1), (694,384), (771,1), (799,1), (811,2083), (823,1), (897,1), (1113,3), (1114,3), (1156,1), (1492,1), (1821,1), (1942,1941)

For any point P not on a sideline of triangle ABC, let \(P ^{-1}\) denote the isogonal conjugate of P. Easily verified properties of eigentransforms include the following.

1. ET(U) = U-Ceva conjugate of \(U ^{-1}\).

2. ET(U) = X₁ if and only if U = X₁ or U lies on the Steiner circumellipse: yz + zx + xy = 0.

3. ET(U) = eigencenter of the anticevian triangle of \(U ^{-1}\).

4. ET(U) is on the self-isogonal cubic Z(U). That is, if \(x : y : z\) denotes a trilinear variable point, then ET(U) is on the cubic

\(ux(y^{2}\) - \(z^{2})\) + \(vy(z^{2}\) - \(x^{2})\) + \(wz(x^{2}\) - \(y^{2})\) = 0.

5. The U-Ceva conjugate of every X on Z(U) is on Z(U); thus, the U-Ceva conjugate, V, of X₁, lies on Z(U); trilinears for \(V ^{-1}\) are

1/(1/v + 1/w - 1/u) : 1/(1/w + 1/u - 1/v) : 1/(1/u + 1/v - 1/w);

ET(U) is the third point of intersection of the line X₁-to-\(V ^{-1}\) and the cubic Z(U).

6. As U is the pivot of Z(U), the following points on Z(U) are collinear: U, ET(U), [ET(U)] ^-1 .

For discussions and generalizations using barycentric coordinates, see Section 1.4 of “Special Isocubics in the Triangle Plane,” downloadable from Bernard Gibert’s Cubics in the Triangle Plane. The cubic Z(U) is there denoted by pK(X6,U), and ET(U) is = (cevian quotient of U and \(U^{*})\) = (U-Ceva conjugate of \(U ^{-1})\); ET(U) is the tangential of \(U^{*}\) in pK(X6,U).

This section was added to ETC on 10/(14-17)/03.