X₃₀₅₇

INTERSECTION OF LINES X₁X_3 AND X₁₀X_11

X₃₀₅₇ = cevapoint of X₁ and X₂₉₄₃

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

X₃₀₅₇ (INTERSECTION OF LINES X₁X_3 AND X₁₀X_11) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}, {10, 11}

• Related Centers: X₁₄₇₆ (-BETH CONJUGATE OF X₁₂₂₂), X₁ (INCENTER), X₃ (CIRCUMCENTER), X₁₀ (SPIEKER CENTER), X₁₁ (FEUERBACH POINT)

Trilinears: \((b + c - a)(b^{2} + c^{2} - 2bc + ab + ac) : :\)

Search Value (6, 9, 13): \(2.850890705409\)

▸ Alternate Coordinates3 forms

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

• Trilinears: \(2 - \cos B - \cos C : :\)

• Trilinears: \(b(\tan B/2) + c(\tan C/2) : :\)

▸ Linear Combinations1 combination

• X₃₀₅₇ = \((R -r)\) X₁ + \(r\) X₃

▸ Incident Central Lines28 lines

{1, 3}, {4, 1000}, {8, 210}, {10, 11}, {12, 946}, {19, 2256}, {21, 643}, {33, 1829}, {34, 1902}, {37, 1953}, {45, 374}, {72, 519}, {140, 1387}, {144, 145}, {190, 1222}, {200, 2136}, {219, 2264}, {220, 2082}, {227, 1457}, {278, 1888}, {355, 1479}, {388, 962}, {496, 1737}, {595, 2361}, {614, 1616}, {1212, 1334}, {1317, 1364}, {2809, 3021}

▸ Functional Relations21 relations

• X₃₀₅₇ = reflection of X_i in X_j for these (i,j): (8, 960), (65, 1)

• X₃₀₅₇ = isogonal conjugate of X₁₄₇₆

• X₃₀₅₇ = complement of X₁₄₉₂₃

• X₃₀₅₇ = X_i-Ceva conjugate of X_j for these (i,j): (1, 1201), (190, 650), (1293, 513)

• X₃₀₅₇ = cevapoint of X₁ and X₂₉₄₃

• X₃₀₅₇ = crosspoint of X₁ and X₈

• X₃₀₅₇ = crosssum of X_i and X_j for these (i,j): (56, 269), (1, 56), (55, 1743)

• X₃₀₅₇ = {X₁,X₄₀}-harmonic conjugate of X₅₆

• X₃₀₅₇ = inverse-in-Feuerbach-hyperbola of X₁₀

• X₃₀₅₇ = X₁ of Mandart-incircle triangle

• X₃₀₅₇ = homothetic center of intangents triangle and reflection of extangents triangle in X₁

• X₃₀₅₇ = bicentric sum of PU(59)

• X₃₀₅₇ = PU(59)-harmonic conjugate of X₆₅₀

• X₃₀₅₇ = {X₁,X₃}-harmonic conjugate of X₁₃₁₉

• X₃₀₅₇ = {X₁,X₆₅}-harmonic conjugate of X₃₅₄

• X₃₀₅₇ = X₂₀ of intouch triangle

• X₃₀₅₇ = X₈ of X₁-Brocard triangle

• X₃₀₅₇ = perspector of Mandart-incircle triangle and Hutson intouch triangle

• X₃₀₅₇ = outer-Garcia-to-inner-Garcia similarity image of X₁₁

• X₃₀₅₇ = X₁₈₈₅-of-excentral-triangle

• X₃₀₅₇ = excentral-to-ABC barycentric image of X₈

Notes

Let O_a be the circle centered at the A-excenter and passing trough the A-intouch point. Define O_b and O_c cyclically. The radical center of O_a, O_b, O_c is X₃₀₅₇. (Randy Hutson, February 10, 2016)

X₃₀₅₇ divides segment X₁X₃ in the ratio r/(R-r). (Randy Hutson, February 10, 2016)

X₃₀₅₇ is the radical center of circles {O_a}, {O_b}, {O_c} used in the construction of the Ursa-minor and Ursa-major triangles; see preamble before X₁₇₆₀₃. (Randy Hutson, June 27, 2018)

Let J_aJ_bJ_c be the excentral triangle. X₃₀₅₇ is the radical center of the 2nd Droz-Farny circles of triangles J_aBC, J_bCA, J_cAB. (Randy Hutson, June 27, 2018)