X1386

MIDPOINT OF INCENTER AND SYMMEDIAN POINT

X1386 is the midpoint of incenter and symmedian point.
< 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, 6 }: Incenter–Symmedian Line { 56, 77 }: Central Line L(56, 77) { 65, 82 }: Central Line L(65, 82) { 81, 105 }: Central Line L(81, 105) { 182, 517 }: Central Line L(182, 517) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_77 = ISOGONAL CONJUGATE OF X(33) (0.5270, 0.1305) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_82 = ISOGONAL CONJUGATE OF X(38) (0.5943, 0.2703) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9714, -0.0933) X_182 = MIDPOINT OF BROCARD DIAMETER (0.5500, 0.1846) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8180, 0.7154) X_1386 = MIDPOINT OF INCENTER AND SYMMEDIAN POINT (0.5857, 0.2503) A B C X1 X6 X56 X77 X65 X82 X81 X105 X182 X517 X1386 (a, b, c) = (5, 6, 7) | AB = 1

X1386 (MIDPOINT OF INCENTER AND SYMMEDIAN POINT) 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 \(f(a,b,c) = a(a + b + c) + (a^{2} + b^{2} + c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(1.341608502261\)
▸ Alternate Coordinates1 form

  • Trilinears: \(as + S_{\omega } : bs + S_{\omega } : cs + S_{\omega }\)

▸ Incident Central Lines15 lines

{1, 6}, {7, 1456}, {56, 77}, {65, 82}, {81, 105}, {141, 1125}, {171, 1054}, {182, 517}, {206, 942}, {241, 1471}, {511, 1385}, {519, 597}, {524, 551}, {614, 940}, {751, 1319}

▸ Functional Relations6 relations