X₃₈₅

X₂-HIRST INVERSE OF X₆

X₃₈₅ is the X₂-hirst inverse of X₆.

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

X₃₈₅ (X₂-HIRST INVERSE OF X₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {32, 76}

• Related Centers: X₆₉₄ (ISOGONAL CONJUGATE OF X₃₈₅), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₃₂ (3rd POWER POINT), X₇₆ (3rd BROCARD POINT)

Trilinears: \(bc(a^{4} - b^{2}c^{2}) : :\)

Barycentrics: \(\cot B + \cot C - \cot A - \tan \omega : :\)

Search Value (6, 9, 13): \(-7.120496928832\)

X₃₈₅ lies on the anti-Brocard circle, the anti-McCay circumcircle.

▸ Incident Central Lines291 lines

{1, 257}, {2, 6}, {3, 194}, {4, 7823}, {5, 7762}, {15, 5980}, {16, 5981}, {20, 6392}, {21, 1655}, {22, 3164}, {23, 523}, {25, 2998}, {30, 148}, {32, 76}, {39, 1078}, {41, 17033}, {55, 192}, {56, 330}, {58, 17499}, {61, 22702}, {62, 22701}, {75, 4386}, {83, 3934}, {98, 511}, {99, 187}, {100, 17759}, {110, 9418}, {111, 892}, {112, 2868}, {114, 5965}, {115, 316}, {140, 13571}, {147, 1513}, {171, 894}, {172, 1909}, {182, 22712}, {190, 17735}, {232, 648}, {237, 3511}, {238, 17793}, {239, 1429}, {248, 290}, {251, 308}, {262, 576}, {264, 10311}, {274, 5277}, {287, 8779}, {297, 9473}, {310, 16956}, {315, 3767}, {335, 3509}, {339, 10317}, {340, 6103}, {350, 1914}, {383, 20426}, {419, 16985}, {512, 9879}, {513, 5990}, {514, 5991}, {518, 5985}, {532, 5979}, {533, 5978}, {543, 6781}, {546, 13111}, {574, 7757}, {575, 15819}, {609, 3761}, {614, 18194}, {620, 7799}, {623, 22511}, {624, 22510}, {625, 7809}, {626, 7755}, {664, 17966}, {668, 5291}, {671, 3849}, {694, 8840}, {698, 2076}, {706, 14871}, {710, 8783}, {730, 11364}, {732, 1691}, {740, 1281}, {858, 16315}, {895, 9769}, {903, 17969}, {1003, 1384}, {1080, 20425}, {1154, 15093}, {1160, 10845}, {1161, 10846}, {1194, 1799}, {1204, 9289}, {1235, 10312}, {1284, 17493}, {1285, 14033}, {1333, 3770}, {1351, 13860}, {1352, 9753}, {1383, 9462}, {1423, 2319}, {1492, 18262}, {1503, 5984}, {1506, 7838}, {1580, 4039}, {1627, 8024}, {1692, 10352}, {1757, 4518}, {1911, 3510}, {1975, 3053}, {2021, 15483}, {2023, 5111}, {2030, 5182}, {2080, 2782}, {2280, 17027}, {2329, 17752}, {2373, 9091}, {2393, 5986}, {2452, 9832}, {2459, 8317}, {2460, 8316}, {2482, 14148}, {2548, 7921}, {2549, 7833}, {2793, 10787}, {2854, 5987}, {2896, 5305}, {2975, 21226}, {2996, 3146}, {3091, 9748}, {3095, 10104}, {3096, 5346}, {3225, 3229}, {3226, 17962}, {3266, 4590}, {3304, 19318}, {3506, 19558}, {3507, 7166}, {3508, 7167}, {3550, 3729}, {3705, 17363}, {3747, 19580}, {3750, 17319}, {3760, 7031}, {3785, 5286}, {3788, 7796}, {3818, 9993}, {3830, 19569}, {3920, 4093}, {3926, 7891}, {3933, 7807}, {3978, 8623}, {3985, 6651}, {3996, 20056}, {4037, 4760}, {4045, 5355}, {4095, 17741}, {4195, 19761}, {4251, 17034}, {4352, 22267}, {4369, 17212}, {4426, 6376}, {4713, 21793}, {4754, 17103}, {4831, 6163}, {5008, 7804}, {5017, 18906}, {5023, 20105}, {5033, 9764}, {5041, 6683}, {5103, 9478}, {5104, 5939}, {5133, 8878}, {5152, 5162}, {5171, 11257}, {5188, 12203}, {5206, 7781}, {5210, 8716}, {5254, 6655}, {5309, 7761}, {5319, 7800}, {5368, 6292}, {5475, 7812}, {5503, 8587}, {5977, 8682}, {5988, 17770}, {5992, 17768}, {6031, 6322}, {6248, 12110}, {6309, 10131}, {6566, 8307}, {6567, 8306}, {6636, 8266}, {6653, 21956}, {6680, 7794}, {6795, 15915}, {7000, 12221}, {7075, 22370}, {7179, 17364}, {7191, 18170}, {7200, 7267}, {7374, 12222}, {7379, 20077}, {7426, 7665}, {7470, 9821}, {7697, 10796}, {7737, 11185}, {7745, 16044}, {7746, 7752}, {7747, 14042}, {7748, 7802}, {7749, 7764}, {7758, 7763}, {7765, 7830}, {7770, 7787}, {7772, 7786}, {7773, 7900}, {7775, 7926}, {7776, 7887}, {7784, 7851}, {7795, 7892}, {7801, 7835}, {7808, 7878}, {7814, 7862}, {7817, 7848}, {7818, 7844}, {7819, 10583}, {7821, 7882}, {7822, 7846}, {7825, 7860}, {7841, 7898}, {7849, 7852}, {7861, 7873}, {7865, 7884}, {7866, 7879}, {7867, 7896}, {7869, 7930}, {7870, 7908}, {7871, 7888}, {7872, 7910}, {7874, 7895}, {7881, 7945}, {7889, 16896}, {7902, 7918}, {7914, 7943}, {8300, 17031}, {8356, 15048}, {8370, 18907}, {8591, 8598}, {8592, 8593}, {8669, 17760}, {9477, 14970}, {9605, 11285}, {9751, 20190}, {9756, 11477}, {9772, 12177}, {9773, 9877}, {9867, 13926}, {9868, 13873}, {10063, 10802}, {10079, 10801}, {10256, 10303}, {10353, 13196}, {10488, 10811}, {10807, 11317}, {10989, 16092}, {11057, 11648}, {11286, 21309}, {12194, 12263}, {13449, 14639}, {14651, 15980}, {14693, 15561}, {14910, 18372}, {16508, 22561}, {16589, 16912}, {16678, 17148}, {16800, 21257}, {16916, 18135}, {16918, 18140}, {16957, 18152}, {17037, 18287}, {17737, 20553}, {17961, 18825}, {18144, 19670}, {18901, 19585}, {18902, 19571}, {18993, 19090}, {18994, 19089}, {20060, 20102}, {20179, 21264}, {20794, 20885}

▸ Functional Relations24 relations

• X₃₈₅ = reflection of X_i in X_j for these (i,j): (99, 187), (147, 1513), (298, 395), (299, 396), (316, 115), (325, 230)

• X₃₈₅ = isogonal conjugate of X₆₉₄

• X₃₈₅ = isotomic conjugate of X₁₉₁₆

• X₃₈₅ = complement of X₇₇₇₉

• X₃₈₅ = anticomplement of X₃₂₅

• X₃₈₅ = circumcircle-inverse of X₃₂₅₃₁

• X₃₈₅ = X_i-Ceva conjugate of X_j for these (i,j): (98, 2), (511, 401), (694, 384)

• X₃₈₅ = crosspoint of X₂₉₀ and X₃₀₈

• X₃₈₅ = crosssum of X_i in X_j for these (i,j): (141, 698), (384, 385)

• X₃₈₅ = crossdifference of every pair of points on line X₃₉X₅₁₂

• X₃₈₅ = trilinear pole of PU(185)

• X₃₈₅ = orthoptic-circle-of-Steiner-inellipe-inverse of X₃₂₅₂₅

• X₃₈₅ = X_i-Hirst inverse of X_j for these (i,j): (2, 6), (3, 194), (171, 894)

• X₃₈₅ = {X₂,X₁₉₃}-harmonic conjugate of X₇₇₇₄

• X₃₈₅ = intersection of trilinear polars of P(1) and U(1)

• X₃₈₅ = trilinear pole of line X₈₀₄X₄₁₀₇ (line is perspectrix of any pair of {ABC, 1st Brocard triangle, 3rd Brocard triangle}, and is also the Lemoine axis of the 1st Brocard triangle.)

• X₃₈₅ = anticomplementary isotomic conjugate of X₁₄₇

• X₃₈₅ = crosspoint of X₆ and X₃₅₁₁ wrt both the excentral and tangential triangles

• X₃₈₅ = inverse-in-Steiner-circumellipse of X₆

• X₃₈₅ = X₉₉-of-1st-anti-Brocard-triangle

• X₃₈₅ = X₉₈-of-anti-McCay-triangle

• X₃₈₅ = barycentric product of PU(133)

• X₃₈₅ = barycentric product X₂₃₉*X₈₉₄

• X₃₈₅ = perspector of ABC and side-triangle of cevian triangles of PU(1)

Notes

Contributed by John Horton Conway, 1998.

X₃₈₅ is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(1) and U(1) of bicentric points (see the notes just before X₁₉₀₈. (Randy Hutson, 9/23/2011)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₈₅. (Randy Hutson, November 30, 2015)

Let L_a be the line through A parallel to the Lemoine axis, and define L_b and L_c cyclically. Let M_a be the reflection of BC in L_a, and define M_b and M_c cyclically. Let A' = M_b∩M_c, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the Lemoine axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X₃₂₅ and centroid X₃₈₅; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, November 30, 2015)