X1116

CENTER OF THE LESTER CIRCLE

The appearance of i in the following list means that Xi lies on the Lester circle: 3, 5, 13, 14, 1117, 5671, 14854, 15475, 15535, 15536, 15537, 15538, 15539, 15540, 15541, 15542, 15543, 15544, 15545, 15546, 15547, 15548, 15549, 15550, 15551, 15552, 15553, 15554, 15555, 34365
< 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) { 115, 125 }: Central Line L(115, 125) { 140, 523 }: Central Line L(140, 523) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_115 = CENTER OF KIEPERT HYPERBOLA (0.7507, 0.0838) X_125 = CENTER OF JERABEK HYPERBOLA (0.6803, 0.0260) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_1116 = CENTER OF THE LESTER CIRCLE (0.7561, 0.0883) A B C X115 X125 X140 X523 X1116 (a, b, c) = (5, 6, 7) | AB = 1

X1116 (CENTER OF THE LESTER CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc(b^{2}-c^{2})[2(a^{2}-b^{2})(c^{2}-a^{2}) + 3R^{2}(2a^{2}-b^{2}-c^{2}) - a^{2}(a^{2}+b^{2}+c^{2}) + a^{4}+b^{4}+c^{4}]\), where R = (a csc A)/2 = circumradius of ABC.
Barycentrics: \((b^{2} - c^{2}) \cdot (-2 a^{8} + 5 a^{6} b^{2} - 3 a^{4} b^{4} - a^{2} b^{6} + b^{8} + 5 a^{6} c^{2} - 8 a^{4} b^{2} c^{2} + 4 a^{2} b^{4} c^{2} - 4 b^{6} c^{2} - 3 a^{4} c^{4} + 4 a^{2} b^{2} c^{4} + 6 b^{4} c^{4} - a^{2} c^{6} - 4 b^{2} c^{6} + c^{8}) : :\)
Search Value (6, 9, 13): \(3.342887310512\)
▸ Incident Central Lines14 lines

{3, 15475}, {4, 39606}, {5, 15543}, {115, 125}, {140, 523}, {381, 32478}, {512, 5892}, {1510, 13363}, {3566, 39482}, {5664, 23105}, {6644, 39481}, {20184, 20299}, {38609, 38611}, {39504, 39512}

▸ Functional Relations6 relations

  • X1116 = 2 X15543 + X18308

  • X1116 = midpoint of Xi and Xj for these {i,j}: {3, 15475}, {5, 15543}, {5664, 23105}

  • X1116 = reflection of X18308 in X5

  • X1116 = tripolar centroid of X13582

  • X1116 = crossdifference of every pair of points on line {110, 11063}

  • X1116 = pole wrt orthocentroidal circle of Napoleon axis (line X6X17)

Notes

The Lester circle passes through the points X3, X5, X13, X14. Coordinates of the center were determined by Milorad Stevanovic (#5895, 9/20/02). The circle is described in

June Lester, “Triangles III: complex centre functions and Ceva’s theorem,” Aequationes Mathematicae 53 (1997) 4-35.