X3560

SS(sin A → cos A) OF X940

X3560 = complement of X6850
< 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) { 2, 3 }: Euler Line { 1, 90 }: Central Line L(1, 90) { 55, 355 }: Central Line L(55, 355) { 119, 498 }: Central Line L(119, 498) { 65, 920 }: Central Line L(65, 920) { 517, 958 }: Central Line L(517, 958) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1 = INCENTER (0.5714, 0.2333) X_90 = -CROSS CONJUGATE OF X(1) (0.1022, 0.0201) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_355 = FUHRMANN CENTER (0.5408, 0.2645) X_119 = FEUERBACH ANTIPODE (0.3776, 0.4312) X_498 = YFF CONCURRENT CONGRUENT CIRCLES POINT (0.5518, 0.2140) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_920 = -CEVA CONJUGATE OF X(1) (0.8006, 0.5344) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8180, 0.7154) X_958 = INTERSECTION OF LINES X(1)X(6) AND X(2)X(12) (0.5166, 0.1682) X_3560 = SS(sin A → cos A) OF X(940) (0.5457, 0.2216) A B C X2 X3 X4 X1 X90 X55 X355 X119 X498 X65 X920 X517 X958 X3560 (a, b, c) = (5, 6, 7) | AB = 1

X3560 (SS(sin A → cos A) OF X940) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \cos^{2} A + 2 \cos B \cos C + \cos C \cos A + \cos A \cos B\)
Search Value (6, 9, 13): \(1.190159839084\)
▸ Alternate Coordinates1 form

  • Trilinears: \(a^{6} - a^{5} (b + c) - 2 a^{4} (b^{2} - b c + c^{2}) + 2 a^{3} (b^{3} + c^{3}) + a^{2} (b^{2} + c^{2})^{2} - a (b^{5} - b^{4} c - b c^{4} + c^{5}) - 2 b c (b^{2} - c^{2})^{2} : :\)

▸ Incident Central Lines49 lines

{1, 90}, {2, 3}, {35, 5587}, {40, 5251}, {55, 355}, {56, 5886}, {58, 5707}, {65, 920}, {84, 5436}, {100, 5818}, {104, 3616}, {119, 498}, {255, 2654}, {284, 5778}, {329, 5761}, {499, 1470}, {515, 5248}, {517, 958}, {573, 4877}, {581, 4653}, {938, 5770}, {943, 4313}, {944, 1621}, {946, 993}, {952, 3295}, {954, 5779}, {956, 1482}, {971, 1001}, {999, 3485}, {1125, 5450}, {1259, 3419}, {1698, 2077}, {1780, 5398}, {1898, 2646}, {2975, 5603}, {3576, 5259}, {3601, 5720}, {3746, 5881}, {3817, 5267}, {3929, 5258}, {4267, 5788}, {4654, 5563}, {5086, 5687}, {5260, 5657}, {5446, 5752}, {5698, 5762}, {5703, 5811}, {5744, 5804}, {5841, 6585}

▸ Functional Relations2 relations

Notes

As a point on the Euler line, X3560 has Shinagawa coefficients \((\$aS_{A}\$, 2abc - \$aS_{A}\$), and also Shinagawa coefficients (R + r, R - r)\).