X511–X565

Centers on the Line at Infinity & Vector Directions

Points on the line at infinity x + y + z = 0 (X511–X526, isogonal conjugates of circumcircle points), unit vector direction centers, and classical midpoints.
< 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 infinity_direction_circle: Infinity direction circle 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_2 = CENTROID (0.5374, 0.2000) X_511 = ISOGONAL CONJUGATE OF X(98) (0.2335, -0.3020) X_514 = ISOGONAL CONJUGATE OF X(101) (1.0761, -0.0329) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.8698, 0.6836) X_528 = DIRECTION OF VECTOR AX + BX + CX, where X = X(11) (1.0237, -0.1286) X_533 = DIRECTION OF VECTOR AX + BX + CX, where X = X(18) (0.0625, -0.1448) X_534 = DIRECTION OF VECTOR AX + BX + CX, where X = X(19) (0.7500, 0.7470) X_537 = DIRECTION OF VECTOR AX + BX + CX, where X = X(38) (-0.0190, 0.3866) X_538 = DIRECTION OF VECTOR AX + BX + CX, where X = X(39) (1.0053, 0.5542) X_539 = DIRECTION OF VECTOR AX + BX + CX, where X = X(54) (1.0917, 0.3927) X_541 = DIRECTION OF VECTOR AX + BX + CX, where X = X(74) (0.7497, -0.3472) X_542 = DIRECTION OF VECTOR AX + BX + CX, where X = X(98) (0.1649, 0.6534) X_543 = DIRECTION OF VECTOR AX + BX + CX, where X = X(99) (0.0220, 0.4806) X_552 = POINT MAIA I (0.6112, 0.3127) X_556 = (X(1),X(75))-ANSWER TO QUESTION A (0.4842, 0.1501) X_564 = INTERSECTION OF LINES X(1)X(1048) AND X(47,91) (0.4993, 0.3100) X_565 = INTERSECTION OF LINES X(49)X(93) AND X(143,324) (0.3278, 0.5134) A B C X3 X1 X2 X511 X514 X527 X528 X533 X534 X537 X538 X539 X541 X542 X543 X552 X556 X564 X565 (a, b, c) = (5, 6, 7) | AB = 1

X511–X565 (Centers on the Line at Infinity & Vector Directions) — 16 representative centers (from 55 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Centers on the line at infinity: X511-X526 (Centers on the lines at infinity: X511-X526)

In barycentric coordinates, the line at infinity, L, is the set of points X =\(x : y : z\) satisfying \(x + y + z = 0\). The isogonal conjugate of L, given by a^2 y z + b^2 z x + c^2 x y = 0, is the circumcircle (e.g., X98-X112).

The collection of collinearities reported below for each of the centers on L comprises a family of parallel lines.

Barycentrics \(x: y : z\) (and trilinears) for X30, X511-X526, and many other triangle centers on L, are polynomials in (a,b,c). For example, barycentrics for X511 are \(p(a,b,c) : p(b,c,a) : p(c,a,b)\), where

p(a,b,c) = \(a^{2}(a^{2}b^{2} + a^{2}c^{2} - b^{4} - c^{4})\).

Such polynomials comprise a family that can be partitioned into two subfamilies: even and odd. Each subfamily can be generated as linear combination of polynomials in certain bases. For example, a basis for the even polynomials of degrees 1 to 4 follows:

degree 1: 2a - b - c
degree 2: \(2a^{2}\) - \(b^{2}\) - \(c^{2}\), 2bc - ca - ab
degree 3: \(2a^{3}\) - \(b^{3}\) - \(c^{3}\), \(a^{2}\)b + \(a^{2}\)c - \(b^{2}\)c - \(b^{2}\)c, \(a^{2}\)b + \(a^{2}\)c - \(b^{2}\)c - \(b^{2}\)c
degree 4: \(2a^{4}\) - \(b^{4}\) - \(c^{4}\), \(a^{3}\)b + \(a^{3}\)c - \(b^{3}\)c - \(b^{3}\)c, \(a^{3}\)b + \(a^{3}\)c - \(b^{3}\)c - \(b^{3}\)c, \(2b^{2}\)\(c^{2}\) - \(a^{2}\)\(b^{2}\) - \(a^{2}\)\(c^{2}\), \(2a^{2}\)bc - \(ab^{2}\)c - \(abc^{2}\)

A basis for the odd polynomials of degrees 1 to 2 follows:

degree 1: b - c
degree 2: (b - c)(a + b + c), (b - c)a

See Clark Kimberling, “A combinatorial classification of triangle centers on the line at infinity,” Journal of Integer Sequences 22 (2019) Article 19.5.4.

Group Preamble — Centers X527-X565 (Centers X527-X565. Jan 1 2001)

Centers X527-X565 were added to ETC on the first day of 2001.

X_515 — ISOGONAL CONJUGATE OF X_102

X515ISOGONAL CONJUGATE OF X102

X515 is the perspector of triangle ABC and the tangential triangles of the conic that passes through the points A, B, C, X2, and X8.
X_516 — ISOGONAL CONJUGATE OF X_103

X516ISOGONAL CONJUGATE OF X103

X516 is the perspector of triangle ABC and the tangential triangles of the conic that passes through the points A, B, C, X2, and X7.
X_530 — DIRECTION OF VECTOR AX + BX + CX, where X = X_13

X530DIRECTION OF VECTOR AX + BX + CX, where X = X13

Let A' be the nine-point center of BCX14, and define B' and C' cyclically. A'B'C' is an equilateral triangle concyclic with X2 and X14 (inscribed in the 2nd Hutson circle). The isogonal (and isotomic) conjugate of X2 wrt A'B'C' is X530. (Randy Hutson, February 10, 2016)
X_531 — DIRECTION OF VECTOR AX + BX + CX, where X = X_14

X531DIRECTION OF VECTOR AX + BX + CX, where X = X14

Let A' be the nine-point center of BCX13, and define B' and C' cyclically. A'B'C' is an equilateral triangle concyclic with X2 and X13 (inscribed in the 1st Hutson circle). The isogonal (and isotomic) conjugate of X2 wrt A'B'C' is X531. (Randy Hutson, February 10, 2016)