X₁₃₄₁

EXSIMILICENTER(CIRCUMCIRCLE, BROCARD CIRCLE)

X₁₃₄₁ is the exsimilicenter(circumcircle, Brocard circle).

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₃₄₁ (EXSIMILICENTER(CIRCUMCIRCLE, BROCARD CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {2, 1348}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂ (CENTROID), X₁₃₄₈ (INSIMILICENTER)

Trilinears: \(a - 2(|OK| - R) \cot \omega \cos A : :\)

Barycentrics: \(a^{2} (a^{4} - a^{2} b^{2} - a^{2} c^{2} - 2 b^{2} c^{2} + (-a^{2} + b^{2} + c^{2}) \sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}) : :\)

Search Value (6, 9, 13): \(1.878637383561\)

▸ Alternate Coordinates1 form

• Trilinears: \(e \cos A - \cos(A - \omega ), : :\), where \(e = (1 - 4 \sin^{2}\omega )^{1/2}\)

▸ Linear Combinations2 combinations

• X₁₃₄₁ = \((-1 + |OK|/R)\) X₃ - X₆

• X₁₃₄₁ = \(2\) X₃ + X₃₅₅₈, X₄ - \(4\) X₁₄₆₃₂, \(4\) X₃₉ - X₃₅₅₇, \(2\) X₃₉ + X₁₃₃₂₆, X₁₃₇₉ + \(2\) X₂₀₂₈, X₃₅₅₇ + \(2\) X₁₃₃₂₆, X₁₃₃₂₅ - \(4\) X₁₃₃₃₄

X₁₃₄₁ lies on the cubics K280, K309, K657, K792, K793, K795, K889, K890, K891, K911.

▸ Incident Central Lines32 lines

{1, 1705}, {2, 1348}, {3, 6}, {4, 1349}, {20, 2543}, {51, 21032}, {55, 1674}, {56, 1675}, {76, 6177}, {110, 6141}, {111, 6142}, {165, 1704}, {262, 6040}, {353, 5639}, {549, 39022}, {958, 1678}, {1083, 11651}, {1344, 2469}, {1345, 2470}, {1376, 1679}, {1503, 19659}, {2039, 7790}, {2549, 31862}, {3414, 7709}, {3589, 19660}, {4045, 14502}, {5091, 36735}, {6189, 7757}, {6190, 7771}, {11650, 11652}, {15048, 39023}, {21036, 22352}

▸ Functional Relations13 relations

• X₁₃₄₁ = external center of similitude of circumcircle and Brocard circle (Peter J. C. Moses, 4/2003)

• X₁₃₄₁ = reflection of X₁₃₄₀ in X₁₁₁₇₁

• X₁₃₄₁ = isogonal conjugate of X₄₆₀₂₄

• X₁₃₄₁ = Brocard-circle-inverse of X₁₃₇₉

• X₁₃₄₁ = 2nd-Brocard-circle-inverse of X₃₅₅₇

• X₁₃₄₁ = Schoutte-circle-inverse of X₁₃₄₀

• X₁₃₄₁ = psi-transform of X₅₆₃₈

• X₁₃₄₁ = crossdifference of every pair of points on line {523, 5639}

• X₁₃₄₁ = intersection of Brocard axis and major axis of Steiner circumellipse

• X₁₃₄₁ = homothetic center of 1st Brocard triangle and circumcevian triangle of X₃₄₁₃

• X₁₃₄₁ = barycentric product X₇₉₉₈*X₄₆₀₂₃

• X₁₃₄₁ = barycentric quotient X₆/X₄₆₀₂₄

• X₁₃₄₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 2542, 1348}, {3, 6, 1379}, {3, 182, 1340}, {3, 14631, 3557}, {3, 38597, 3098}, {6, 574, 1340}, {6, 1379, 3557}, {6, 2028, 3558}, {15, 16, 1340}, {32, 5116, 1340}, {39, 2028, 6}, {39, 3094, 1340}, {39, 13326, 3557}, {371, 372, 14630}, {566, 2088, 1340}, {1342, 1343, 2559}, {1379, 14631, 6}, {1670, 1671, 3557}, {1689, 1690, 1340}, {2028, 8589, 44453}, {2080, 39498, 1340}, {3102, 3103, 13325}, {3106, 3107, 1340}, {5013, 5028, 1340}, {5050, 9734, 1340}, {5092, 26316, 1340}, {8589, 10485, 1340}, {13326, 14631, 3558}, {14899, 35607, 31863}, {15815, 39764, 1340}, {30260, 30261, 1340}, {39162, 39163, 5638}