X3581

INTERSECTION OF LINES X3X_6 AND X30X_74

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A* be the crosssum of A1 and A2, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X3581.
< 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) { 3, 6 }: Brocard Axis { 30, 74 }: Central Line L(30, 74) { 186, 323 }: Central Line L(186, 323) { 49, 1658 }: Central Line L(49, 1658) { 399, 1495 }: Central Line L(399, 1495) { 381, 1531 }: Central Line L(381, 1531) 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_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_186 = INVERSE-IN-CIRCUMCIRCLE OF X(4) (0.7957, 0.8758) X_323 = ISOTOMIC CONJUGATE OF X(94) (-0.8839, -1.3275) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_1658 = CIRCUMCENTER OF KOSNITA TRIANGLE (0.3191, -0.3713) X_399 = PARRY REFLECTION POINT (0.0032, 0.9936) X_1495 = CROSSSUM OF X(2) AND X(30) (0.8194, 1.4855) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_1531 = ORTHOJOIN OF X(32) (0.0445, -0.5419) X_3581 = INTERSECTION OF LINES X(3)X(6) AND X(30)X(74) (1.6355, 1.9774) A B C X3 X6 X30 X74 X186 X323 X49 X1658 X399 X1495 X381 X1531 X3581 (a, b, c) = (5, 6, 7) | AB = 1

X3581 (INTERSECTION OF LINES X3X_6 AND X30X_74) 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 \(f(a,b,c) = a[2b^{4} + 2c^{4} - a^{4} - a^{2}b^{2} - a^{2}c^{2} - 4b^{2}c^{2}][(b^{2} + c^{2} - a^{2})^{2} - b^{2}c^{2}]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(2.821233402219\)
▸ Incident Central Lines9 lines

{3, 6}, {30, 74}, {49, 1658}, {185, 2937}, {186, 323}, {381, 1531}, {399, 1495}, {541, 1533}, {1204, 1657}

▸ Functional Relations2 relations

  • X3581 = reflection of Xi in Xj for these (i,j): (265, 3580), (323, 1511), (399, 1495)

  • X3581 = perspector of ABC and cross-triangle of ABC and 2nd isogonal triangle of X4