X1282

5th SHARYGIN POINT

X1282 is the perspector of the excentral triangle and the triangle D'E'F' constructed at X1281. Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.
< 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, 41 }: Central Line L(1, 41) { 43, 57 }: Central Line L(43, 57) { 10, 150 }: Central Line L(10, 150) { 63, 100 }: Central Line L(63, 100) { 40, 170 }: Central Line L(40, 170) { 152, 516 }: Central Line L(152, 516) 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_41 = -CEVA CONJUGATE OF X(31) (0.5823, 0.2244) X_43 = -CEVA CONJUGATE OF X(1) (0.6358, 0.2963) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_150 = REFLECTION OF X(20) IN X(103) (1.1837, -0.4499) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_40 = BEVAN POINT (0.4286, -0.0292) X_170 = -ALEPH CONJUGATE OF X(9) (0.4465, -0.0557) X_152 = REFLECTION OF X(20) IN X(101) (0.0408, 1.2414) X_516 = ISOGONAL CONJUGATE OF X(103) (0.7703, 0.7386) X_1282 = 5th SHARYGIN POINT (-0.1429, 0.8165) A B C X1 X41 X43 X57 X10 X150 X63 X100 X40 X170 X152 X516 X1282 (a, b, c) = (5, 6, 7) | AB = 1

X1282 (5th SHARYGIN 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 \(f(a,b,c) = a^{4} + a^{3}(b + c) - a^{2}(2b^{2} + 3bc + 2c^{2})\), + a(b + c)(\(b^{2}\) + \(c^{2})\) - (b - c)2(\(b^{2}\) + bc + \(c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.950704963246\)

X1282 lies on the Bevan circle.

▸ Incident Central Lines11 lines

{1, 41}, {8, 1281}, {10, 150}, {40, 170}, {43, 57}, {55, 846}, {63, 100}, {152, 516}, {354, 1051}, {518, 910}, {659, 926}

▸ Functional Relations1 relation

Notes

Let JaJbJc be the excentral triangle. Let La be the line parallel to the Brocard axis of BCJa and passing through Ja. Define Lb and Lc cyclically. The lines La, Lb, Lc concur in X1282. (Randy Hutson, December 2, 2017)