X2446–X2520

Circle & Conic Intersections, Lemoine Inverses & Radical Centers I

Intersections of central lines with the incircle, Bevan circle, Steiner inellipse, and Steiner circumellipse, inverses in the 1st and 2nd Lemoine circles, and circle radical centers (X2446–X2520).
< 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_2446 = 1st INTERSECTION(LINE X(1)X(3), INCIRCLE) (0.4599, 0.0284) X_2463 = INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, INCIRCLE) (0.5738, 0.2778) X_2468 = EXSIMILICENTER(ORTHOCENTROIDALCIRCLE, SPIEKER CIRCLE) (1.0566, 1.3119) X_2472 = EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, 1st LEMOINE CIRCLE) (0.5889, 0.3622) X_2481 = INTERSECTION(FEUERBACH HYPERBOLA, STEINER CIRCUMELLIPSE) (0.9225, -0.0635) X_2482 = STEINER-INELLIPSE-ANTIPODE OF X(115) (0.3242, 0.3161) X_2486 = RADICAL CENTER OF {INCIRCLE, NINE-POINT CIRCLE, MOSES CIRCLE} (0.7431, 0.0749) X_2489 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, 2ND LEMOINE CIRCLE} (0.1726, 0.9764) X_2491 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT-CIRCLE, GALLATLY CIRCLE} (1.0687, 0.6339) X_2492 = RADICAL CENTER OF {CIRCUMCIRCLE,NINE-POINT-CIRCLE, MOSES CIRCLE} (1.1810, 0.5910) X_2497 = RADICAL CENTER OF {CIRCUMCIRCLE, INCIRCLE, GALLATLY CIRCLE} (0.6018, 0.9166) X_2501 = RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT CIRCLE, TAYLOR CIRCLE} (0.4430, 0.3272) X_2504 = RADICAL CENTER OF {INCIRCLE, NINE-POINT CIRCLE, BROCARD CIRCLE} (0.4545, -0.2079) X_2507 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, GALLATLY CIRCLE} (0.8163, 0.3364) X_2508 = RADICAL CENTER OF {NINE-POINT CIRCLE, 1st LEMOINE CIRCLE, MOSES CIRCLE} (0.9539, 0.3233) X_2519 = RADICAL CENTER OF {NINE-POINT, 2nd LEMOINE, BROCARD} CIRCLES (0.6756, -0.2289) A B C X3 X1 X2446 X2463 X2468 X2472 X2481 X2482 X2486 X2489 X2491 X2492 X2497 X2501 X2504 X2507 X2508 X2519 (a, b, c) = (5, 6, 7) | AB = 1

X2446–X2520 (Circle & Conic Intersections, Lemoine Inverses & Radical Centers I) — 16 representative centers (from 75 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — (Mostly) Circle-related Points, X2446 - X2573

Centers X2446 to X2573 were contributed by Peter J. C. Moses during October to November, 2003. See the notes just before X1662 for an introduction and notation.

Included in this section are centers of similitude of several pairs of circles. However, for some pairs of central circles, the two centers of similitudes are not triangle centers but are instead a pair of bicentric points. Specifically, Moses found that the insimilicenter (i.e., internal center of similitude) and exsimilicenter of the following pairs are bicentric points; as such they are listed in the Bicentric Pairs section (accessed by the Tables button atop this page):

{Parry circle and 2nd Lemoine circle}; see P(62) in Bicentric Pairs
{Parry circle and circumcircle}; P(63)
{Parry circle and incircle}; P(64)
{Parry circle and nine-point circle}; P(65)
{Parry circle and Spieker circle}; P(66)
{Parry circle and Moses circle}; P(67)

The 2nd Brocard circle, defined in the aforementioned Bicentric Pairs, is mentioned several times [from X2554 to X2573]. A summary of its properties follows.

Definition: The 2nd Brocard circle is the circle having center X3 and radius eR, where e = (1 - 4 \(\sin^{2}\)ω)1/2 and R = circumradius. The 2nd Brocard circle and the Brocard circle meet in the 1st and 2nd Brocard points. Peter J. C. Moses found that the 2nd Brocard circle also passes through the points Xi for I = 1670, 1671, 2554, 2555, 2556, 2557. The square of the radius of the 2nd Brocard circle can be written out as

\(a^{2}\)\(b^{2}\)\(c^{2}(a^{4} + b^{4} + c ^{4} - b^{2}c^{2} - c^{2}a^{2} - a^{2}b^{2})\)/[4(\(b^{2}\)\(c^{2}\) + \(c^{2}\)\(a^{2}\) + \(a^{2}\)\(b^{2})\)(\(a^{2}\) + \(b^{2}\) + \(c^{2})\)]

Xi is the inverse-in-the-2nd-Brocard-circle of Xj for these (i,j): (6,39), (76,99), (1666,2563), (1667,2562), (1689,1690), (2026,2561), (2027,2560)

The power of the 2nd Brocard circle with respect to each of the vertices A, B, C is \(a^{2}\)\(b^{2}\)\(c^{2}\)/(\(b^{2}\)\(c^{2}\) + \(c^{2}\)\(a^{2}\) + \(a^{2}\)\(b^{2})\)