X3119–X3217

Danneels Perspectors & Xi-Ceva Conjugates

Eric Danneels’ parallel-cevian perspectors (X3119–X3126) and a systematic catalog of Xi-Ceva conjugates of classical and central points (X3127–X3217).
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_3119 = DANNEELS PERSPECTOR FOR X(9) (0.6663, 0.0328) X_3120 = DANNEELS PERSPECTOR FOR X(10) (0.7602, 0.0916) X_3126 = DANNEELS PERSPECTOR FOR X(100) (0.2802, 0.2871) X_3127 = -CEVA CONJUGATE OF X(1162) (0.6030, 0.3716) X_3131 = -CEVA CONJUGATE OF X(6) (0.5395, 0.2055) X_3132 = -CEVA CONJUGATE OF X(6) (0.4024, -0.1533) X_3133 = -CEVA CONJUGATE OF X(52) (0.6612, 0.5237) X_3140 = -CEVA CONJUGATE OF X(523) (0.3136, -0.3857) X_3141 = -CEVA CONJUGATE OF X(523) (0.2422, -0.5725) X_3142 = -CEVA CONJUGATE OF X(523) (0.5719, 0.2902) X_3143 = -CEVA CONJUGATE OF X(523) (0.2819, -0.4686) X_3146 = -CEVA CONJUGATE OF X(2) (0.8367, 0.9831) X_3147 = -CEVA CONJUGATE OF X(4) (0.9224, 1.2074) X_3149 = -CEVA CONJUGATE OF X(1) (0.7068, 0.6431) X_3150 = -CEVA CONJUGATE OF X(525) (0.4495, -0.0300) X_3154 = -CEVA CONJUGATE OF X(523) (0.4976, 0.0958) A B C X3 X1 X3119 X3120 X3126 X3127 X3131 X3132 X3133 X3140 X3141 X3142 X3143 X3146 X3147 X3149 X3150 X3154 (a, b, c) = (5, 6, 7) | AB = 1

X3119–X3217 (Danneels Perspectors & Xi-Ceva Conjugates) — 16 representative centers (from 99 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Danneels Perspectors

Suppose A1B1C1 is the cevian triangle of a point X.
Let LAB be the line through B parallel to A1B1, and let LAC be the line through C parallel to A1C1.
Let A2 = LAB∩LAC, and define B2 and C2 cyclically. Let

A3 = BB2∩CC2, B3 = CC2∩AA2, C3 = AA2∩BB2.

Eric Danneels proves in “A Simple Perspectivity,” :cite:`Forum Geometricorum <forum-geometricorum>` 6 (2006) 199-203, that the triangles A3B3C3 and ABC are perspective. If X =\(x : y : z\) (barycentrics), then the Danneels perspector P(X) is given by

P(X) = x(y - z)2 : y(z - x)2 : z(x - y)2.

If X' denotes the isotomic conjugate of X, then P(X') = P(X). A few examples selected from Danneels’s article follow:

P(X1) = P(X75) = X244
P(X3) = P(X264) = X2972
P(X4) = P(X69) = X125
P(X7) = P(X8) = X11.
X_3160 — X_2-CEVA CONJUGATE OF X_7

X3160-CEVA CONJUGATE OF X7

X3160 = perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C and has center X7
X_3161 — X_2-CEVA CONJUGATE OF X_8

X3161-CEVA CONJUGATE OF X8

X3161 = perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C and has center X8