X₁₃₄₀

INSIMILICENTER(CIRCUMCIRCLE, BROCARD CIRCLE)

X₁₃₄₀ is the insimilicenter(circumcircle, Brocard circle).

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

X₁₃₄₀ (INSIMILICENTER(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, 1349}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂ (CENTROID), X₁₃₄₉ (EXSIMILICENTER)

Trilinears: \(a + 2(|OK| + R) \cot \omega \cos A : :\), where |OK| = distance between X₃ and X₆, ω = Brocard angle of ABC, and R = circumradius of ABC

Search Value (6, 9, 13): \(4.728910741187\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations2 combinations

• X₁₃₄₀ = \((1 + |OK|/R)\) X₃ + X₆

• X₁₃₄₀ = \(2\) X₃ + X₃₅₅₇, X₄ - \(4\) X₁₄₆₃₃, \(4\) X₃₉ - X₃₅₅₈, \(2\) X₃₉ + X₁₃₃₂₅, X₁₃₈₀ + \(2\) X₂₀₂₉, X₃₅₅₈ + \(2\) X₁₃₃₂₅, X₁₃₃₂₆ - \(4\) X₁₃₃₃₄

X₁₃₄₀ lies on the cubics K280, K309, K657, K792, K793, K795, K889, K890, K891, K911.

▸ Incident Central Lines34 lines

{1, 1704}, {2, 1349}, {3, 6}, {4, 1348}, {20, 2542}, {51, 21036}, {55, 1675}, {56, 1674}, {76, 6178}, {110, 6142}, {111, 6141}, {165, 1705}, {262, 6039}, {353, 5638}, {549, 39023}, {958, 1679}, {1083, 11652}, {1344, 2470}, {1345, 2469}, {1376, 1678}, {1503, 19660}, {2039, 43461}, {2040, 7790}, {2549, 31863}, {3413, 7709}, {3589, 19659}, {3972, 46023}, {4045, 14501}, {5091, 36736}, {6189, 7771}, {6190, 7757}, {11650, 11651}, {15048, 39022}, {21032, 22352}

▸ Functional Relations15 relations

• X₁₃₄₀ = reflection of X₁₃₄₁ in X₁₁₁₇₁

• X₁₃₄₀ = isogonal conjugate of X₄₆₀₂₃

• X₁₃₄₀ = Brocard-circle-inverse of X₁₃₈₀

• X₁₃₄₀ = Schoutte-circle-inverse of X₁₃₄₁

• X₁₃₄₀ = 2nd-Brocard-circle-inverse of X₃₅₅₈

• X₁₃₄₀ = psi-transform of X₅₆₃₉

• X₁₃₄₀ = crossdifference of every pair of points on line {523, 5638}

• X₁₃₄₀ = internal center of similitude of circumcircle and Brocard circle (Peter J. C. Moses, 4/2003)

• X₁₃₄₀ = {X₃,X₁₈₂}-harmonic conjugate of X₁₃₄₁

• X₁₃₄₀ = radical center of Lucas(t) circles, for t where circles are tangent to Brocard circle

• X₁₃₄₀ = intersection of Brocard axis and minor axis of Steiner circumellipse

• X₁₃₄₀ = homothetic center of 1st Brocard triangle and circumcevian triangle of X₃₄₁₄

• X₁₃₄₀ = barycentric product X₇₉₉₈*X₄₆₀₂₄

• X₁₃₄₀ = barycentric quotient X₆/X₄₆₀₂₃

• X₁₃₄₀ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 2543, 1349}, {3, 6, 1380}, {3, 182, 1341}, {3, 14630, 3558}, {3, 38596, 3098}, {6, 574, 1341}, {6, 1380, 3558}, {6, 2029, 3557}, {15, 16, 1341}, {32, 5116, 1341}, {39, 2029, 6}, {39, 3094, 1341}, {39, 13325, 3558}, {371, 372, 14631}, {566, 2088, 1341}, {1342, 1343, 2558}, {1380, 14630, 6}, {1670, 1671, 3558}, {1689, 1690, 1341}, {2029, 8589, 44453}, {2080, 39498, 1341}, {3102, 3103, 13326}, {3106, 3107, 1341}, {5013, 5028, 1341}, {5050, 9734, 1341}, {5092, 26316, 1341}, {8589, 10485, 1341}, {13325, 14630, 3557}, {15815, 39764, 1341}, {30260, 30261, 1341}, {35608, 35609, 31862}