X31–X58

Power Points, Circle Similicenters & Named Centers

2nd and 3rd power points, the Bevan point, Brocard midpoint, Kosnita point, and internal and external centers of similitude of the circumcircle, incircle, and nine-point circle.
< 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_31 = 2nd POWER POINT (0.6228, 0.3008) X_33 = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES (0.6019, 0.3546) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_39 = BROCARD MIDPOINT (0.5778, 0.2305) X_40 = BEVAN POINT (0.4286, -0.0292) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_53 = SYMMEDIAN POINT OF ORTHIC TRIANGLE (0.6164, 0.4390) A B C X3 X1 X31 X33 X35 X36 X39 X40 X49 X52 X53 (a, b, c) = (5, 6, 7) | AB = 1

X31–X58 (Power Points, Circle Similicenters & Named Centers) — 9 representative centers (from 28 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_34 — X_4-BETH CONJUGATE OF X_4

X34-BETH CONJUGATE OF X4

X34 is the center of perspective of the orthic triangle and the reflection in the incenter of the intangents triangle.
X_39 — BROCARD MIDPOINT

X39BROCARD MIDPOINT

X39 is the midpoint of the 1st and 2nd Brocard points, given by trilinears \(c/b : a/c : b/a\) and \(b/c : c/a : a/b\). The third and fourth trilinear representations were given by Peter J. C. Moses (10/3/03); cf. X511, X32, X182.
X_40 — BEVAN POINT

X40BEVAN POINT

X40 = point of concurrence of the perpendiculars from the excenters to the respective sides
X_43 — X_6-CEVA CONJUGATE OF X_1

X43-CEVA CONJUGATE OF X1

X43 is the {X2,X42}-harmonic conjugate of X1. X43 is the external center of similitude of the Bevan circle and Apollonius circle; the internal center is X1695.
X_47 — X_110-BETH CONJUGATE OF X_34

X47-BETH CONJUGATE OF X34

X47 = perspector of ABC and extraversion triangle of X47 (which is also the anticevian triangle of X47)
X_49 — CENTER OF SINE-TRIPLE-ANGLE CIRCLE

X49CENTER OF SINE-TRIPLE-ANGLE CIRCLE

Suppose that P and Q are distinct points in the plane of a triangle ABC . Let PA = reflection of P in line AQ, let QA = reflection of Q in line AP, and let MA = midpoint of segment PAQA. Define MB and MC cyclically. César Lozada found that if Q = isogonal conjugate of P, then the locus of P for which MAMBMC is perspective to ABC is the union of a cubic and 6 circles: specifically, the McCay cubic (K003), the circles {B,C,Jb,Jc}}, {C,A,Jc,Ja}}, {A,B,Ja,Jb}}, and the circles {B,C,Ja}}, {C,A,Jb}}, {A,B,Jc}}, where Ja,Jb,Jc are the excenters of ABC. Moreover, if P = X3 and Q = X4, then MAMBMC is not only perspective, but homothetic, to ABC, and the center of homothety is X49. See Hyacinthos 23265, June 1, 2015.
X_54 — KOSNITA POINT

X54KOSNITA POINT

Let O be the circumcenter of triangle ABC, and OA the circumcenter of triangle BOC. Define OB and OC cyclically. Then the lines AOA, BOB, COC concur in X54.