X1662–X1706

Circle-Related Points

Internal and external centers of similitude among the circumcircle, incircle, nine-point circle, Brocard circle, 1st and 2nd Lemoine circles, Apollonius circle, and Bevan 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_1662 = 1st INTERSECTION OF BROCARD AXIS AND 1st LEMOINE CIRCLE (0.4002, -0.0628) X_1666 = 1st INTERSECTION OF BROCARD AXIS AND 2nd LEMOINE CIRCLE (0.4587, 0.0339) X_1667 = 2nd INTERSECTION OF BROCARD AXIS AND 2nd LEMOINE CIRCLE (0.7413, 0.5005) X_1668 = EXSIMILICENTER(BROCARD CIRCLE, 2nd LEMOINE CIRCLE) (0.5226, 0.1394) X_1673 = EXSIMILICENTER(INCIRCLE, 1st LEMOINE CIRCLE) (0.6607, 0.4358) X_1676 = INSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE) (0.5533, 0.2188) X_1677 = EXSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE) (0.6019, 0.7290) X_1679 = EXSIMILICENTER(BROCARD CIRCLE, SPIEKER CIRCLE) (0.6921, 0.1911) X_1681 = EXSIMILICENTER(1st LEMOINE CIRCLE, SPIEKER CIRCLE) (0.5004, 0.1824) X_1684 = EXSIMILICENTER(1st LEMOINE CIRCLE, APOLLONIUS CIRCLE) (0.6037, 0.2733) X_1686 = EXSIMILICENTER(2nd LEMOINE CIRCLE, APOLLONIUS CIRCLE) (0.6588, 0.3643) X_1688 = EXSIMILICENTER(1st LEMOINE CIRCLE, 2nd LEMOINE CIRCLE) (1.4224, 1.6255) X_1691 = RADICAL TRACE OF CIRCUMCIRCLE AND 1st LEMOINE CIRCLE (0.9991, 0.9263) X_1692 = RADICAL TRACE OF CIRCUMCIRCLE AND 2nd LEMOINE CIRCLE (0.7995, 0.5968) X_1695 = INSIMILICENTER(BEVAN CIRCLE, APOLLONIUS CIRCLE) (0.3697, -0.1216) X_1699 = EXSIMILICENTER(BEVAN CIRCLE, NINE-POINT CIRCLE) (0.5986, 0.3416) A B C X3 X1 X1662 X1666 X1667 X1668 X1673 X1676 X1677 X1679 X1681 X1684 X1686 X1688 X1691 X1692 X1695 X1699 (a, b, c) = (5, 6, 7) | AB = 1

X1662–X1706 (Circle-Related Points) — 16 representative centers (from 45 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Circle-related Points 1662-1706

The next cluster of points were contributed by Peter J. C. Moses during August, 2003. Symbols used to represent functions of a,b,c (or A, B, C) include

σ = area of triangle ABC
ω = arccot(cot A + cot B + cot C) = arccot[(\(a^{2}\) + \(b^{2}\) +\(c^{2})\)/(4σ)]; ω is the Brocard angle of ABC
e = \(\sqrt{1 - 4 \sin^{2}\omega }\) (as in Gallatly, p. 96, along with other formulas involving ω)
s = (a + b + c)/2 = semiperimeter of ABC
r = σ/s = inradius of ABC

Circles mentioned in this section are the following:

Name Center Radius circumcircle X3 R incircle X1 r nine-point circle X5 R/2 Brocard circle X182 eR/(2 cos ω) 1st Lemoine circle X182 (1/2)(R sec ω) 2nd Lemoine circle (cosine circle) X6 abc/(\(a^{2}\) + \(b^{2}\) + \(c^{2})\) Spieker circle X10 r/2 Apollonius circle X970 (\(r^{2}\) + \(s^{2})\)/4r Bevan circle X40 2R

For further information on many circles, click MathWorld and scroll down to links to various specific circles.

X_1682 — INSIMILICENTER(INCIRCLE, APOLLONIUS CIRCLE)

X1682INSIMILICENTER(INCIRCLE, APOLLONIUS CIRCLE)

The exsimilicenter of the incircle and Apollonius circle is X181. Also, the triangle A'B'C' formed (as at X2092 by the intersections of the Apollonius circle and the excircles is perspective to the cevian triangle of X1, and the perspector is X1682. (Paul Yiu, Hyacinthos #8076, 10/01/03)