X2077

INVERSE-IN-CIRCUMCIRCLE OF X40

X2077 = reflection of X36 in X3
< 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) { 36, 3 }: OI Line (Incenter–Circumcenter Line) { 1, 3 }: OI Line (Incenter–Circumcenter Line) { 30, 119 }: Central Line L(30, 119) { 100, 515 }: Central Line L(100, 515) { 104, 519 }: Central Line L(104, 519) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_119 = FEUERBACH ANTIPODE (0.3776, 0.4312) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_515 = ISOGONAL CONJUGATE OF X(102) (0.6804, 0.7691) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_519 = ISOGONAL CONJUGATE OF X(106) (0.9566, 0.6107) X_2077 = INVERSE-IN-CIRCUMCIRCLE OF X(40) (-0.3333, -1.4289) A B C X36 X3 X1 X30 X119 X100 X515 X104 X519 X2077 (a, b, c) = (5, 6, 7) | AB = 1

X2077 (INVERSE-IN-CIRCUMCIRCLE OF X40) 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 \(f(a,b,c) = r + 2(r - R) \cos A\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(16.140012575052\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \cos 2A + \cos B + \cos C + (2 \cos B + 2 \cos C - 3) \cos A\)

▸ Incident Central Lines19 lines

{1, 3}, {24, 1753}, {30, 119}, {78, 1158}, {84, 1259}, {100, 515}, {102, 901}, {104, 519}, {122, 856}, {376, 535}, {386, 601}, {404, 946}, {516, 1519}, {912, 1768}, {953, 1293}, {972, 1308}, {1012, 1376}, {1593, 1878}, {1618, 1818}

▸ Functional Relations5 relations

Notes

Let P and Q be circumcircle antipodes. X2077 is the line X1X3 intercept, other than X1, of circle {{X1,P,Q}}. (Randy Hutson, August 28, 2020)