X560

4th POWER POINT

X560 = crosssum of X75 and X304
< 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) { 31, 48 }: Central Line L(31, 48) { 1, 82 }: Central Line L(1, 82) { 42, 584 }: Central Line L(42, 584) { 100, 697 }: Central Line L(100, 697) { 101, 713 }: Central Line L(101, 713) { 110, 715 }: Central Line L(110, 715) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_561 = ISOGONAL CONJUGATE OF 4th POWER POINT (0.4126, 0.1125) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5412, 0.1246) X_1 = INCENTER (0.5714, 0.2333) X_82 = ISOGONAL CONJUGATE OF X(38) (0.5943, 0.2703) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_584 = ORTHOHARMONIC OF X(500) (0.5720, 0.2209) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_697 = EVEN (- 4, - 3) CIRCUMCIRCLE POINT (0.9705, 0.2996) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.2143, 0.5249) X_713 = EVEN (- 3, - 2) CIRCUMCIRCLE POINT (0.9916, 0.2389) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_715 = EVEN (- 3, - 1) CIRCUMCIRCLE POINT (1.0000, 0.2040) X_560 = 4th POWER POINT (0.6520, 0.3639) A B C X561 X31 X48 X1 X82 X42 X584 X100 X697 X101 X713 X110 X715 X560 (a, b, c) = (5, 6, 7) | AB = 1

X560 (4th POWER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a^{4} : b^{4} : c^{4}\)
Barycentrics: \(a^{5} : b^{5} : c^{5}\)
Search Value (6, 9, 13): \(0.140003184510\)
▸ Alternate Coordinates4 forms

  • Trilinears: \(a^{2}(S_{A} - S_{\omega }) : :\)

  • Trilinears: \((\cot B + \cot C)^{2} : :\)

  • Trilinears: \((S_{B} + S_{C})^{2} : :\)

  • Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = \((distance from A to de Longchamps line)^{2}\)

▸ Incident Central Lines9 lines

{1, 82}, {31, 48}, {41, 872}, {42, 584}, {100, 697}, {101, 713}, {110, 715}, {717, 825}, {719, 827}

▸ Functional Relations9 relations

Notes

X560 is the vertex conjugate of the foci of the inellipse that is the trilinear square of the Lemoine axis. The center of the inellipse is X16584 and its Brianchon point (perspector) is X31. (Randy Hutson, October 15, 2018)