X2001–X2054

Orthocorrespondents, Gallatly Circle & Euler-Vecten-Gibert Points

Orthocorrespondents of classical centers (X2001–X2006), similicenters and intersections of the Gallatly circle (X2007–X2040), and the six Euler-Vecten-Gibert points and harmonic conjugates (X2041–X2054).
< 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 nine_point_circle: Nine-Point Circle (Euler / Feuerbach Circle) { 2, 3 }: Euler Line 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_5 = NINE-POINT CENTER (0.5561, 0.2489) X_2 = CENTROID (0.5374, 0.2000) X_2001 = ORTHOCORRESPONDENT OF X(32) (0.4984, 0.2475) X_2003 = ORTHOCORRESPONDENT OF X(36) (0.5923, 0.2428) X_2006 = ORTHOCORRESPONDENT OF X(80) (0.9679, -0.1714) X_2008 = EXSIMILICENTER(GALLATLY CIRCLE, INCIRCLE) (0.3669, 0.3221) X_2010 = EXSIMILICENTER(GALLATLY CIRCLE, NINE-POINT CIRCLE) (0.9341, -0.0718) X_2012 = EXSIMILICENTER(GALLATLY CIRCLE, BROCARD CIRCLE) (0.5314, 0.1539) X_2016 = EXSIMILICENTER(GALLATLY CIRCLE, ORTHOCENTROIDAL CIRCLE) (0.5725, 0.3498) X_2021 = RADICAL TRACE OF GALLATLY CIRCLE AND CIRCUMCIRCLE (0.8882, 0.7432) X_2022 = RADICAL TRACE OF GALLATLY AND BROCARD CIRCLES (0.3296, -0.1794) X_2023 = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES (0.6642, 0.1572) X_2024 = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES (0.6887, 0.4136) X_2026 = 1st BROCARD-AXIS-GALLATLY-CIRCLE INTERSECTION (0.4532, 0.0248) X_2037 = 1st BROCARD-AXIS-APOLLONIUS-CIRCLE INTERSECTION (1.2120, 1.2780) X_2041 = 1st EULER-VECTEN-GIBERT POINT (0.9189, 1.1981) X_2043 = 3rd EULER-VECTEN-GIBERT POINT (0.4352, -0.0675) X_2052 = ISOGONAL CONJUGATE OF X(577) (0.6161, 0.4941) A B C X3 X1 X5 X2 X2001 X2003 X2006 X2008 X2010 X2012 X2016 X2021 X2022 X2023 X2024 X2026 X2037 X2041 X2043 X2052 (a, b, c) = (5, 6, 7) | AB = 1

X2001–X2054 (Orthocorrespondents, Gallatly Circle & Euler-Vecten-Gibert Points) — 16 representative centers (from 54 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Gallatly Circle, etc., 2007-2040

In his book The Modern Geometry of the Triangle, 2nd edition (Francis Hodgson, London, 1913), William Gallatly, on page 117, introduces the pedal cicle of the 1st and 2nd Brocard points. The circle is here named the Gallatly circle. Its center is the Brocard midpoint, X39, and the radius, R sin ω, where ω denotes the Brocard angle, given by

tan ω = 4[area(ABC)]/(\(a^{2}\) + \(b^{2}\) + \(c^{2})\).

Other special symbols used in this section are identified just before X1662.

Centers X2007-X2040 were contributed with coordinates by Peter J. C. Moses during September, 2003. For centers of similitude of the Gallatly circle and the circumcircle, see X1689 and X1690; for those of the Gallatly circle and the 2nd Lemoine circle, see X1671 and X1670.

Group Preamble — Euler-Vecten-Gibert Points, 2041 -2046

On August 13, 2003, Bernard Gibert contributed six centers that lie on the Euler line and are related to the Vecten points, X485 and X486.

X_2023 — RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES

X2023RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES

Let A'B'C' be the 1st Brocard triangle. Let A″ be the orthogonal projection of A on line B'C', and define B″ and C″ cyclically. Triangle A″B″C″ is perpsective to the medial triangle at X2023. (Note that the A-vertex of the medial triangle is the orthogonal projection of A' on line BC, and cyclically for B' and C'.) (Randy Hutson, July 11, 2019)
X_2028 — 1st BROCARD-AXIS-MOSES-CIRCLE INTERSECTION

X20281st BROCARD-AXIS-MOSES-CIRCLE INTERSECTION

Of the two points of intersection of the Brocard axis, X3X6 with the Moses circle, X2028 is the one nearer to X3. Points X2028 and X2029 are the points of intersection of the Brocard axis and the asymptotes of the Kiepert hyperbola; see X2039 and X2040. For a description of the Moses circle and others, see the notes just above X1662.
X_2051 — EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE)

X2051EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE)

X2051 is the external center of similitude of the nine-point and Apollonius circles. Trilinears for the internal center, X10, result from g(a,b,c) by changing the “-” just after “cos A″ to “+”. (The two circles are described just before X1662.)