X₄₁₉

X₄-HIRST INVERSE OF X₂₅

X₄₁₉ is the X₄-hirst inverse of X₂₅.

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

X₄₁₉ (X₄-HIRST INVERSE OF X₂₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {238, 242}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₂₃₈ (-LINE CONJUGATE OF X₃₇), X₂₄₂ (-LINE CONJUGATE OF X₇₁)

Trilinears: \((u^{2} - vw)\sec A : (v^{2} - wu)\sec B : (v^{2} - uv)\sec C\), where \(u : v : w\) = X₃₁; e.g., \(u = u(A,B,C) = a^{2}\)

Barycentrics: \((u^{2} - vw)\tan A : (v^{2} - wu)\tan B : (w^{2} - uv)\tan C\)

Search Value (6, 9, 13): \(0.834756041461\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a^{2} - b c) \cdot (a^{2} + b c) \cdot (a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) :\)

X₄₁₉ lies on the cubics K252 and K785.

▸ Incident Central Lines80 lines

{2, 3}, {6, 3186}, {76, 9306}, {83, 3866}, {98, 42671}, {99, 19599}, {107, 2698}, {110, 51481}, {154, 39646}, {182, 43976}, {184, 40814}, {206, 41760}, {232, 6531}, {238, 242}, {264, 1974}, {275, 3399}, {290, 9418}, {316, 32223}, {338, 18374}, {385, 16985}, {394, 12251}, {648, 35146}, {685, 1976}, {1215, 1840}, {1235, 38830}, {1301, 48259}, {1304, 43654}, {1495, 41254}, {1576, 44375}, {1632, 3003}, {1843, 36794}, {2001, 44077}, {2052, 3406}, {2201, 19557}, {2207, 3224}, {2211, 41520}, {2501, 3288}, {2679, 40077}, {3167, 7754}, {3168, 8743}, {3229, 33874}, {3563, 6037}, {3978, 12215}, {4369, 22093}, {5254, 10192}, {6375, 51988}, {6530, 34130}, {6749, 41585}, {7760, 34986}, {7762, 41588}, {8623, 51324}, {8754, 37765}, {9307, 46432}, {9308, 19118}, {9407, 9512}, {10345, 11386}, {10359, 10601}, {11064, 43453}, {11596, 40135}, {12243, 35266}, {14602, 40820}, {17907, 41762}, {17941, 47736}, {18020, 34537}, {21445, 35278}, {27377, 41584}, {32085, 41884}, {32713, 37778}, {33630, 34208}, {35282, 38227}, {39089, 44090}, {40146, 43678}, {41194, 44778}, {41195, 44779}, {41253, 44084}, {41259, 51843}, {43696, 43721}, {44142, 46104}, {46151, 52418}, {51246, 51948}

▸ Functional Relations26 relations

• X₄₁₉ = reflection of X₁₁₅₉₆ in X₄₀₁₃₅

• X₄₁₉ = isogonal conjugate of X₃₆₂₁₄

• X₄₁₉ = isotomic conjugate of X₄₀₇₀₈

• X₄₁₉ = orthocentroidal-circle-inverse of X₅₁₁₇

• X₄₁₉ = polar-circle-inverse of X₁₁₀₀₇

• X₄₁₉ = polar conjugate of X₁₉₁₆

• X₄₁₉ = complement of the isogonal conjugate of X₄₃₇₂₁

• X₄₁₉ = isotomic conjugate of the isogonal conjugate of X₄₄₀₈₉

• X₄₁₉ = isogonal conjugate of the isotomic conjugate of X₁₇₉₈₄

• X₄₁₉ = polar conjugate of the isotomic conjugate of X₃₈₅

• X₄₁₉ = polar conjugate of the isogonal conjugate of X₁₆₉₁

• X₄₁₉ = X₄₃₇₂₁-complementary conjugate of X₁₀

• X₄₁₉ = X_i-Ceva conjugate of X_j for these (i,j): {232, 41204}, {6531, 4}, {17984, 385}, {41174, 112}, {47736, 39931}

• X₄₁₉ = X_i-cross conjugate of X_j for these (i,j): {1691, 385}, {39931, 4}

• X₄₁₉ = X_i-isoconjugate of X_j for these (i,j): {1, 36214}, {3, 1581}, {31, 40708}, {48, 1916}, {63, 694}, {69, 1967}, {75, 17970}, {184, 1934}, {256, 295}, {257, 2196}, {291, 7015}, {293, 40810}, {304, 9468}, {305, 1927}, {326, 17980}, {335, 7116}, {336, 14251}, {337, 904}, {647, 37134}, {656, 805}, {810, 18829}, {882, 4592}, {1911, 7019}, {1959, 15391}, {3917, 43763}, {4020, 14970}, {8789, 40364}, {9247, 18896}, {14208, 17938}

• X₄₁₉ = X_i-Dao conjugate of X_j for these (i,j): {2, 40708}, {3, 19576}, {3, 36214}, {48, 39031}, {63, 39043}, {69, 8290}, {132, 40810}, {206, 17970}, {304, 39044}, {325, 6393}, {525, 35078}, {694, 3162}, {805, 40596}, {882, 5139}, {1249, 1916}, {1581, 36103}, {2491, 41172}, {3917, 36213}, {6651, 7019}, {7015, 39029}, {8623, 36212}, {15259, 17980}, {18829, 39062}, {37134, 39052}, {39030, 40364}

• X₄₁₉ = cevapoint of X₁₆₉₁ and X₄₄₀₈₉

• X₄₁₉ = crosspoint of X_i and X_j for these (i,j): {685, 18020}, {16081, 46104}

• X₄₁₉ = crosssum of X_i and X_j for these (i,j): {684, 20975}, {3289, 20775}, {12215, 37894}

• X₄₁₉ = trilinear pole of line {804, 12829}

• X₄₁₉ = X₄-Hirst inverse of X₂₅

• X₄₁₉ = pole wrt polar circle of trilinear polar of X₁₉₁₆ (line X₁₄₁X₅₂₃)

• X₄₁₉ = crossdifference of every pair of points on line {647, 3917}

• X₄₁₉ = barycentric product X_i*X_j for these {i,j}: {4, 385}, {6, 17984}, {19, 1966}, {25, 3978}, {27, 4039}, {76, 44089}, {92, 1580}, {98, 39931}, {107, 24284}, {112, 14295}, {172, 40717}, {230, 47736}, {232, 14382}, {239, 7009}, {242, 894}, {264, 1691}, {290, 51324}, {297, 40820}, {350, 7119}, {393, 12215}, {648, 804}, {732, 32085}, {862, 8033}, {880, 2489}, {1215, 31905}, {1783, 14296}, {1840, 33295}, {1874, 27958}, {1897, 4107}, {1909, 2201}, {1926, 1973}, {1933, 1969}, {1974, 14603}, {2501, 17941}, {2679, 41174}, {3186, 39927}, {4019, 34856}, {4032, 14024}, {4164, 6335}, {5026, 17983}, {5027, 6331}, {5976, 6531}, {8623, 46104}, {12829, 35142}, {14006, 16609}, {14602, 18022}, {16080, 51430}, {16081, 36213}, {16089, 32542}, {16985, 37892}, {18901, 44162}, {18902, 44161}, {36820, 37765}, {39495, 46456}

• X₄₁₉ = barycentric quotient X_i/X_j for these {i,j}: {2, 40708}, {4, 1916}, {6, 36214}, {19, 1581}, {25, 694}, {32, 17970}, {92, 1934}, {112, 805}, {162, 37134}, {172, 295}, {232, 40810}, {239, 7019}, {242, 257}, {264, 18896}, {385, 69}, {458, 8842}, {460, 47734}, {648, 18829}, {685, 39291}, {732, 3933}, {804, 525}, {894, 337}, {1580, 63}, {1691, 3}, {1840, 43534}, {1914, 7015}, {1926, 40364}, {1933, 48}, {1966, 304}, {1973, 1967}, {1974, 9468}, {1976, 15391}, {2086, 20975}, {2201, 256}, {2207, 17980}, {2210, 7116}, {2211, 14251}, {2489, 882}, {2679, 41172}, {3978, 305}, {4027, 12215}, {4039, 306}, {4107, 4025}, {4164, 905}, {5026, 6390}, {5027, 647}, {5976, 6393}, {6531, 36897}, {7009, 335}, {7119, 291}, {7122, 2196}, {8623, 3917}, {11183, 14417}, {11325, 47642}, {12215, 3926}, {12829, 3564}, {14006, 36800}, {14295, 3267}, {14296, 15413}, {14602, 184}, {14603, 40050}, {16985, 37894}, {17941, 4563}, {17980, 41517}, {17984, 76}, {18020, 39292}, {18022, 44160}, {18901, 40360}, {18902, 14575}, {24284, 3265}, {31905, 32010}, {32085, 14970}, {32542, 14941}, {32544, 8858}, {35325, 46161}, {36213, 36212}, {36820, 34897}, {37892, 40847}, {39495, 8552}, {39927, 43714}, {39931, 325}, {40717, 44187}, {40820, 287}, {42396, 41209}, {44089, 6}, {44102, 18872}, {44162, 8789}, {47736, 8781}, {50732, 51454}, {51320, 37893}, {51324, 511}, {51430, 11064}

• X₄₁₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 4, 5117}, {2, 3148, 37334}, {2, 6620, 4}, {2, 33336, 384}, {4, 186, 35474}, {4, 420, 297}, {4, 38282, 52283}, {22, 37190, 7470}, {22, 41238, 37190}, {23, 46571, 14957}, {25, 458, 4}, {25, 11325, 46505}, {29, 37389, 37362}, {237, 401, 11676}, {237, 1316, 401}, {297, 460, 4}, {297, 468, 420}, {421, 468, 450}, {460, 468, 297}, {462, 468, 470}, {463, 468, 471}, {468, 7473, 186}, {2450, 44887, 2}, {3144, 37055, 6998}, {5000, 5001, 5999}, {6660, 21531, 5999}, {14957, 46493, 23}, {14957, 46512, 46571}, {19128, 44145, 41204}, {21177, 46544, 15013}, {21531, 37906, 6660}, {37912, 46522, 15014}, {42789, 42790, 47620}, {44887, 50707, 2450}, {46493, 46512, 14957}

Notes

As a point on the Euler line, X₄₁₉ has Shinagawa coefficients \((2(E + F)F,-(E + F)^{2} + S^{2})\).