X1284

8th SHARYGIN POINT

X1284 is the homothetic center of the intouch intriangle and the triangle DEF 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) { 7, 21 }: Central Line L(7, 21) { 37, 65 }: Central Line L(37, 65) { 1, 256 }: Central Line L(1, 256) { 57, 846 }: Central Line L(57, 846) { 513, 663 }: Central Line L(513, 663) { 350, 1281 }: Central Line L(350, 1281) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_7 = GERGONNE POINT (0.5903, 0.2769) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_1 = INCENTER (0.5714, 0.2333) X_256 = 1st SHARYGIN POINT (0.5601, 0.2146) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_846 = 4th SHARYGIN POINT (0.5437, 0.1969) X_513 = ISOGONAL CONJUGATE OF X(100) (1.0529, -0.0806) X_663 = CROSSDIFFERENCE OF X(2) AND X(7) (8.6667, -3.2660) X_350 = -HIRST INVERSE OF X(75) (-3.5227, -3.1370) X_1281 = 3rd SHARYGIN POINT (0.3110, 0.3100) X_1284 = 8th SHARYGIN POINT (0.9846, 0.9156) A B C X7 X21 X37 X65 X1 X256 X57 X846 X513 X663 X350 X1281 X1284 (a, b, c) = (5, 6, 7) | AB = 1

X1284 (8th 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 \(f(a,b,c) = (b + c)(bc - a^{2})/(b + c - a)\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-0.497494428014\)
▸ Incident Central Lines6 lines

{1, 256}, {7, 21}, {37, 65}, {57, 846}, {350, 1281}, {513, 663}

▸ Functional Relations5 relations