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X₂₃₆₅–X₂₄₄₅¶

Circumcircle CIR(X) Points & Gibert-Simson Transforms

Circumcevian-eigentransform points CIR(X) on the circumcircle and their isogonal conjugates (X₂₃₆₅–X₂₃₉₃), followed by Gibert-Simson transforms GS(X) on the Simson cubic K010 (X₂₃₉₄–X₂₄₁₉) and their isogonal conjugates on K162 (X₂₄₂₀–X₂₄₄₅).

X₂₃₆₅–X₂₄₄₅ (Circumcircle CIR(X) Points & Gibert-Simson Transforms) — 16 representative centers (from 81 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — More Points on the Circumcircle, 2365-2384

Suppose that U =\(u : v : w\) (trilinears) is a triangle center other than X₆, and let CIR(U) be the point given by trilinears

\(F(u,v,w) : F(v,w,u) : F(w,u,v)\) = avw/(\(av^{2}\) + \(aw^{2}\) - buv - cuw) : bwu/(\(bw^{2}\) + \(bu^{2}\) - cvw - avu) : cuv/(\(cu^{2}\) + \(cv^{2}\) - awu - bwv)

CIR(U) lies on the circumcircle. Pairs (i,j) such that X_j = CIR(X_i) include the following:

(1,106), (2,729), (3,1300), (4,1294), (9,1477), (31,767), (37,741), (41,767), (43,106), (44,106), (74,1294), (75,701), (110,99), (125,827), (194,729), (238,741), (265,1300), (1084,689), (1154,1141), (1279,(1477), (1499,1296), (1503,1297), (1510,930), (1634,689)

This section was added to ETC on 10/20/03; the preamble was revised on January 20, 2015, following suggestions by Viktor Kitaysky. Points with trilinears similar to those of CIR(U), called circum-eigentransforms, denoted by CET(U), are given in Section 4.

Group Preamble — GIBERT-SIMSON TRANSFORMS, X₂₃₉₄ -X₂₄₁₉

On October 19, 2003, Bernard Gibert contributed points that lie on the Simson cubic. Indeed, this cubic is the image of the circumcircle under a mapping here named the Gibert-Simson transform. If U =\(u : v : w\) (barycentrics) and U is not X₃, then the transform is given by

GS(U) = (\(b^{2}\)\(S_{B}\)/v - \(c^{2}\)\(S_{C}\)/w)u/\(a^{2}\) : (\(c^{2}\)\(S_{C}\)/w - \(a^{2}\)\(S_{A}\)/u)v/\(b^{2}\) : (\(a^{2}\)\(S_{A}\)/u - \(b^{2}\)\(S_{B}\)/v)w/\(c^{2}\).

In case U lies on the circumcircle, these barycentrics are proportional to the respective directed distances from A, B, C, to the Simson line of U. If U* is the antipode of U, then GS(U*) is the isotomic conjugate of GS(U). In the Catalogue at Cubics in the Triangle Plane, the Simson cubic is discussed as K010, and its isogonal transform is the cubic K162. The isogonal conjugates of centers X₂₃₆₅-X₂₄₁₉ are indexed as X₂₄₂₀-X₂₄₄₅.

If U =\(u : v : w\) =\(u(a,b,c) : u(b,c,a) : u(c,a,b)\) (trilinears), then

GS(U) =\(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (v \cos C - w \cos B)(u/a)^{2}\),

and the cubic K010 is given by trilinears

(cos A)\(x(b^{2}\)\(y^{2}\) + \(c^{2}\)\(z^{2})\) + (cos B)\(y(c^{2}\)\(z^{2}\) + \(a^{2}\)\(x^{2})\) + (cos C)\(z(a^{2}\)\(x^{2}\) + \(b^{2}\)\(x^{2})\) - (\(a^{2}\) + \(b^{2}\) + \(c^{2})\)xyz = 0,

and the cubic K162 by

(cos A)\(x(c^{2}\)\(y^{2}\) + \(b^{2}\)\(z^{2})\) + (cos B)\(y(a^{2}\)\(z^{2}\) + \(c^{2}\)\(x^{2})\) + (cos C)\(z(b^{2}\)\(x^{2}\) + \(a^{2}\)\(x^{2})\) - (\(a^{2}\) + \(b^{2}\) + \(c^{2})\)xyz = 0,

In the definition of eigentransform before X₂₁₂₀, if cevian triangle is replaced by circumcevian triangle, the result is CET(U) as discussed before X₂₃₆₅. Gibert observed that if cevian triangle is replaced by pedal triangle, for U on the circumcircle, the result is the isogonal conjugate of GS(U). That is, [GS(U)] ^-1 is the eigencenter of the pedal triangle of U (which is a degenerate triangle lying in the Simson line of U).

Regarding triangle centers that do not lie on the circumcircle, GS(X_i) = X_j for these (i,j): (32,669), (48,1459), (187,1649), (248,879), (485,850), (486,850).

This section was added to ETC on 10/23/03. For more Gibert-Simson transforms, see X₁₅₄₁₁-X₁₅₄₂₃