X1500

INSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

The circle having center X39 and radius R tan ω sin 2ω, where R denotes the circumradius of triangle ABC, is here introduced as the Moses circle. It is tangent to the nine-point circle at X115, and its internal and external centers of similitude with the incircle are X1500 and X1015, respectively. (Peter J. C. Moses, 5/29/03)
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 1, 39 }: Central Line L(1, 39) { 10, 37 }: Central Line L(10, 37) { 32, 55 }: Central Line L(32, 55) { 12, 115 }: Central Line L(12, 115) { 35, 172 }: Central Line L(35, 172) { 42, 213 }: Central Line L(42, 213) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1509 = ISOGONAL CONJUGATE OF X(1500) (0.5682, 0.2347) X_1 = INCENTER (0.5714, 0.2333) X_39 = BROCARD MIDPOINT (0.5778, 0.2305) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_172 = TRILINEAR PRODUCT X(6)*X(171) (0.6080, 0.2866) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_213 = -CEVA CONJUGATE OF X(42) (0.6135, 0.2833) X_1500 = INSIMILICENTER OF MOSES CIRCLE AND INCIRCLE (0.5747, 0.2319) A B C X1509 X1 X39 X10 X37 X32 X55 X12 X115 X35 X172 X42 X213 X1500 (a, b, c) = (5, 6, 7) | AB = 1

X1500 (INSIMILICENTER OF MOSES CIRCLE AND INCIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a(b + c)^{2} : b(c + a)^{2} : c(a + b)^{2}\)
Barycentrics: \(a^{2}(b + c)^{2} : b^{2}(c + a)^{2} : c^{2}(a + b)^{2}\)
Search Value (6, 9, 13): \(1.622887773302\)
▸ Linear Combinations1 combination

  • X1500 = \((\tan \omega \sin 2\omega )R\) X1 + \(r\) X39

▸ Incident Central Lines18 lines

{1, 39}, {6, 595}, {10, 37}, {11, 1508}, {12, 115}, {32, 55}, {35, 172}, {41, 1017}, {42, 213}, {56, 574}, {76, 192}, {216, 1062}, {346, 941}, {519, 1107}, {612, 1196}, {756, 762}, {1124, 1505}, {1335, 1504}

▸ Functional Relations6 relations