X1994

ORTHOCORRESPONDENT OF X5

Let OAOBOC be the Kosnita triangle. Let A' be the trilinear pole of line OBOC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X1994. (Randy Hutson, March 21, 2019)
< 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, 6 }: Centroid–Symmedian Line { 52, 54 }: Central Line L(52, 54) { 51, 110 }: Central Line L(51, 110) { 49, 143 }: Central Line L(49, 143) { 5, 195 }: Central Line L(5, 195) { 23, 184 }: Central Line L(23, 184) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_2 = CENTROID (0.5374, 0.2000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_54 = KOSNITA POINT (0.5880, 0.2176) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_195 = -CEVA CONJUGATE OF X(3) (0.6761, 0.3331) X_23 = FAR-OUT POINT (1.3871, 2.4232) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_1994 = ORTHOCORRESPONDENT OF X(5) (0.6334, 0.3031) A B C X3 X2 X6 X52 X54 X51 X110 X49 X143 X5 X195 X23 X184 X1994 (a, b, c) = (5, 6, 7) | AB = 1

X1994 (ORTHOCORRESPONDENT OF X5) 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 \(f(a,b,c) = a[b^{2}c^{2} - 16(\operatorname{area}(ABC))^{2}]\)
Barycentrics: \(a^{2} - R^{2} : :\)
Search Value (6, 9, 13): \(-1.238671087458\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\cos 3A \csc 2A : \cos 3B \csc 2B : \cos 3C \csc 2C ( M. Iliev, 4/12/07)\)

▸ Incident Central Lines15 lines

{2, 6}, {3, 1199}, {5, 195}, {22, 1351}, {23, 184}, {49, 143}, {51, 110}, {52, 54}, {94, 275}, {97, 216}, {186, 568}, {427, 1353}, {567, 1154}, {1194, 1570}, {1627, 1692}

▸ Functional Relations8 relations