X2594–X2676

Bicentric Pairs PU(68)–PU(85) & Golden Arbelos Points

Sums, differences, crosssums, crossdifferences, trilinear poles, and midpoints of bicentric pairs PU(68) through PU(85) (X2594–X2670), together with the 1st and 2nd Golden Arbelos points (X2671–X2676).
< 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_2594 = SUM OF PU(68) (0.5791, 0.2254) X_2597 = TRILINEAR POLE OF PU(68) (-0.0331, 0.2426) X_2606 = CROSSSUM OF PU(70) (1.2843, -0.4943) X_2611 = DIFFERENCE OF PU(71) (0.7448, 0.0628) X_2612 = CROSSSUM OF PU(71) (0.8019, 0.3283) X_2613 = CROSSDIFFERENCE OF PU(70) (0.2669, 0.5328) X_2632 = DIFFERENCE OF PU(75) (0.6212, 0.0090) X_2633 = CROSSSUM OF PU(75) (0.7202, 0.5544) X_2640 = CROSSSUM OF PU(78) (1.3971, -0.3206) X_2644 = CROSSSUM OF PU(79) (0.7523, 0.2158) X_2648 = TRILINEAR POLE OF PU(80) (0.9134, -0.0753) X_2656 = TRILINEAR POLE OF PU(82) (0.6337, -0.4192) X_2659 = CROSSDIFFERENCE OF PU(83) (0.7496, 1.2245) X_2660 = TRILINEAR POLE OF PU(83) (0.6083, -0.1529) X_2665 = TRILINEAR POLE OF PU(84) (1.0458, 0.0952) X_2672 = 2nd GOLDEN ARBELOS POINT (0.3508, 0.0923) A B C X3 X1 X2594 X2597 X2606 X2611 X2612 X2613 X2632 X2633 X2640 X2644 X2648 X2656 X2659 X2660 X2665 X2672 (a, b, c) = (5, 6, 7) | AB = 1

X2594–X2676 (Bicentric Pairs PU(68)–PU(85) & Golden Arbelos Points) — 16 representative centers (from 83 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Centers Related to Bicentric Pairs

Most of the centers X2594 - X2670 are defined in terms of bicentric pairs of points (e.g., the 1st and 2nd Brocard points).

For a list of bicentric pairs and associated terminology and properties, press the Tables button at the top of this page.

X_2594 — SUM OF PU(68)

X2594SUM OF PU(68)

The notation “PU(68)”, as described in the Bicentric Pairs section accessible by the Tables button, abbreviates a bicentric pair of points P =\(p : q : r\) and U =\(u : v : w\), where p, q, r have the form g(a,b,c), g(b,c,a), g(c,a,b) and u,v,w have the form h(a,b,c), h(b,c,a), h(c,a,b). “Sum of PU(68)” denotes the point
X_2671 — 1st GOLDEN ARBELOS POINT

X26711st GOLDEN ARBELOS POINT

Suppose UV is a segment and 1/2 < s < 1. Let P be the point on UV satisfying |UP|/|VP| = s. Let O be a semicircle having diameter UV. Let H be the semicircle having diameter VP, on the same side of UV as O. Let K be the semicircle having diameter UP, on the same side of UP as O. The three circular arcs form an arbelos, R.
X_2672 — 2nd GOLDEN ARBELOS POINT

X26722nd GOLDEN ARBELOS POINT

In the construction of X2671, if A' is the center of the s-Archimedean circle of the inward arbelos on segment BC, and B' and C' are defined cyclically, then A'A, B'B, C'C concur in X2672.
X_2675 — X_2671X_2673∩X_2672X_2674

X2675∩X2672X_2674

This point of intersection and the fact that it lies on the Euler line were contributed by Peter J. C. Moses, August, 31, 2004.