X572

ORTHOGONAL OF X58

X572 = inverse-in-Brocard-circle of X573
< 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 { 9, 48 }: Central Line L(9, 48) { 54, 71 }: Central Line L(54, 71) { 51, 199 }: Central Line L(51, 199) { 1, 604 }: Central Line L(1, 604) { 169, 610 }: Central Line L(169, 610) 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_9 = MITTENPUNKT (0.5110, 0.1615) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5412, 0.1246) X_54 = KOSNITA POINT (0.5880, 0.2176) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4807, 0.0915) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_199 = -CEVA CONJUGATE OF X(6) (0.4796, 0.0487) X_1 = INCENTER (0.5714, 0.2333) X_604 = -CEVA CONJUGATE OF X(31) (0.6442, 0.3744) X_169 = -CEVA CONJUGATE OF X(1) (0.5497, 0.2510) X_610 = -CEVA CONJUGATE OF X(1) (0.4310, -0.2713) X_572 = ORTHOGONAL OF X(58) (7.2953, -0.6907) A B C X3 X6 X9 X48 X54 X71 X51 X199 X1 X604 X169 X610 X572 (a, b, c) = (5, 6, 7) | AB = 1

X572 (ORTHOGONAL OF X58) 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 \(f(A,B,C) = \cos(A - U)\), where cot \(U = \cot(A/2) \cot(B/2) \cot(C/2)\) (Joe Goggins, 11/26/08)
Search Value (6, 9, 13): \(5.234901251064\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \sin A + \cot D/2 \cos A\), cot \(D/2 = (a + b + c)^{2}/(4 \operatorname{area})\)

  • Trilinears: \(s \cos A + r \sin A : :\)

  • Trilinears: \(a (a^{3} - a (b^{2} - b c + c^{2}) - b c (b + c)) : :\)

▸ Linear Combinations1 combination

  • X572 = \(s\) X3 + \(r \cot(\omega )\) X6

▸ Incident Central Lines14 lines

{1, 604}, {3, 6}, {9, 48}, {51, 199}, {54, 71}, {103, 825}, {165, 1051}, {169, 610}, {184, 1011}, {219, 947}, {261, 662}, {517, 1100}, {594, 952}, {631, 966}

▸ Functional Relations4 relations

Notes

X572 is the intersection of the Brocard axes of ABC, the Feuerbach triangle, and the Apollonius triangle. (Randy Hutson, March 21, 2019)