X₂₆₂

ISOGONAL CONJUGATE OF X₁₈₂

Let P_a be the parabola with focus A and directrix BC. Let L_a be the polar of X₃ wrt P_a and define L_b and L_c cyclically. Let A' = L_b∩L_c, B' = L_c∩L_a, C' = L_a∩L_b. The lines AA', BB', CC' concur in X₂₆₂. Also, X₂₆₂ is also the isotomic conjugate, wrt A'B'C', of X₃.

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

X₂₆₂ (ISOGONAL CONJUGATE OF X₁₈₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 39}, {2, 51}

• Related Centers: X₁₈₂ (MIDPOINT OF BROCARD DIAMETER), X₄ (ORTHOCENTER), X₃₉ (BROCARD MIDPOINT), X₂ (CENTROID), X₅₁ (CENTROID OF ORTHIC TRIANGLE)

Trilinears: \(\sec(A - \omega ) : \sec(B - \omega ) : \sec(C - \omega )\)

Barycentrics: \(\sin A \sec(A - \omega ) : \sin B \sec(B - \omega ) : \sin C \sec(C - \omega )\)

Search Value (6, 9, 13): \(-0.849220039334\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(1/(a^{4} - a^{2}b^{2} - a^{2}c^{2} - 2b^{2}c^{2}) : :\)

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

▸ Linear Combinations1 combination

• X₂₆₂ = \(4\) X₂ - X₃₃₇₀₆, \(2\) X₃ - \(5\) X₇₇₈₆, X₃ - \(4\) X₁₁₂₇₂, X₃ + \(2\) X₁₄₈₈₁, X₄ + \(2\) X₃₉, \(2\) X₄ + X₁₁₂₅₇, \(4\) X₅ - X₇₆, \(2\) X₅ + X₃₀₉₅, X₂₀ - \(4\) X₁₃₃₃₄, \(4\) X₃₉ - X₁₁₂₅₇, X₇₆ + \(2\) X₃₀₉₅, X₉₈ - \(4\) X₂₀₂₃, \(2\) X₁₁₄ + X₁₉₁₆, \(2\) X₁₁₉ + X₃₂₄₅₄, \(4\) X₁₄₀ - X₉₈₂₁, X₁₉₄ + \(5\) X₃₀₉₁, X₁₉₄ + \(2\) X₆₂₄₈, \(2\) X₃₅₅ + X₇₉₇₆, \(2\) X₃₈₁ + X₇₇₅₇, \(3\) X₃₈₁ - \(2\) X₂₂₆₈₁, \(3\) X₃₈₁ + X₃₂₅₁₉, \(2\) X₅₄₆ + X₃₂₄₄₈, \(5\) X₆₃₁ - \(2\) X₅₁₈₈, \(5\) X₆₃₁ - \(8\) X₆₆₈₃, \(2\) X₉₄₆ + X₁₂₇₈₂, X₁₃₅₀ - \(4\) X₁₀₀₀₇, \(2\) X₁₃₅₂ + X₃₂₄₅₁, \(7\) X₃₀₉₀ - \(4\) X₃₉₃₄, \(7\) X₃₀₉₀ - X₁₂₂₅₁, \(5\) X₃₀₉₁ - \(2\) X₆₂₄₈, X₃₀₉₄ + \(2\) X₅₄₈₀, X₃₁₄₆ + \(5\) X₃₂₅₂₂, \(5\) X₃₆₁₈ - \(2\) X₁₃₃₅₄, X₃₆₂₇ + \(2\) X₃₂₅₁₆, \(4\) X₃₆₂₈ - X₃₂₅₂₁, \(7\) X₃₈₅₁ - X₁₃₁₀₈, \(11\) X₃₈₅₅ + \(4\) X₃₂₄₅₀, \(4\) X₃₉₃₄ - X₁₂₂₅₁, \(11\) X₅₀₅₆ - \(5\) X₃₁₂₇₆, \(8\) X₅₀₆₆ + X₁₁₀₅₅, \(13\) X₅₀₆₇ - \(10\) X₃₁₂₃₉, \(13\) X₅₀₆₈ - X₂₀₀₈₁, \(5\) X₅₀₇₁ - \(2\) X₉₄₆₆, \(11\) X₅₀₇₂ + X₃₂₅₂₀, X₅₁₈₈ - \(4\) X₆₆₈₃, \(4\) X₅₄₇₆ - X₂₂₄₈₆, \(2\) X₆₁₉₄ - \(3\) X₂₂₇₁₂, \(4\) X₆₁₉₄ - \(3\) X₃₃₇₀₆, \(2\) X₆₂₄₉ + X₃₂₄₇₆, X₆₃₀₉ - \(4\) X₇₇₆₄, X₇₇₀₉ + \(2\) X₂₂₆₈₂, \(3\) X₇₇₅₇ + \(4\) X₂₂₆₈₁, \(3\) X₇₇₅₇ - \(2\) X₃₂₅₁₉, X₇₇₅₉ + \(2\) X₃₂₁₈₉, \(2\) X₇₇₇₅ + X₁₃₀₈₅, \(5\) X₇₇₈₆ - \(8\) X₁₁₂₇₂, \(5\) X₇₇₈₆ + \(4\) X₁₄₈₈₁, \(5\) X₇₇₈₆ - X₂₂₆₇₆, \(5\) X₇₇₈₆ + \(2\) X₂₂₇₂₈, X₇₈₂₃ + \(2\) X₃₂₁₅₂, \(5\) X₇₉₂₁ + X₉₈₆₃, \(7\) X₇₉₈₉ - X₉₉₀₂, \(5\) X₈₂₂₇ - \(2\) X₁₂₂₆₃, \(2\) X₈₃₅₆ + X₃₄₇₃₃, \(8\) X₉₃₀₀ + X₁₄₄₅₈, \(7\) X₉₇₈₁ - \(4\) X₂₇₃₇₅, \(2\) X₉₈₈₀ + X₁₁₁₅₂, X₁₁₂₅₇ + \(4\) X₂₂₆₈₂, \(2\) X₁₁₂₇₂ + X₁₄₈₈₁, \(8\) X₁₁₂₇₂ - X₂₂₆₇₆, \(4\) X₁₁₂₇₂ + X₂₂₇₂₈, \(4\) X₁₂₁₀₂ + \(5\) X₃₂₅₂₃, X₁₃₃₂₅ - \(4\) X₁₄₆₃₃, X₁₃₃₂₆ - \(4\) X₁₄₆₃₂, \(2\) X₁₄₂₅₁ + X₁₄₂₆₅, \(3\) X₁₄₅₆₁ - X₃₁₉₅₈, \(3\) X₁₄₆₃₉ + X₃₂₄₆₉, \(4\) X₁₄₈₈₁ + X₂₂₆₇₆, \(4\) X₁₅₈₁₉ - \(3\) X₂₂₇₁₂, \(8\) X₁₅₈₁₉ - \(3\) X₃₃₇₀₆, \(2\) X₂₂₄₇₅ + X₂₂₆₅₀, \(4\) X₂₂₄₇₅ - X₂₂₇₁₃, \(2\) X₂₂₆₅₀ + X₂₂₇₁₃, X₂₂₆₇₆ + \(2\) X₂₂₇₂₈, \(2\) X₂₂₆₈₁ + \(3\) X₃₂₄₄₇, \(2\) X₂₂₆₈₁ + X₃₂₅₁₉, \(3\) X₃₂₄₄₇ - X₃₂₅₁₉

X₂₆₂ lies on the Kiepert circumhyperbola, the cubics K062, K263, K280, K300, K305, K353, K412, K422, K423, K581, K582, K677, K756, K790, K791, K802, K930, K939, K1012.

▸ Incident Central Lines252 lines

{1, 22475}, {2, 51}, {3, 83}, {4, 39}, {5, 76}, {6, 98}, {10, 2186}, {11, 12837}, {12, 12836}, {13, 383}, {14, 1080}, {17, 3104}, {18, 3105}, {20, 5395}, {23, 7578}, {25, 275}, {30, 598}, {32, 3406}, {54, 3202}, {55, 22556}, {56, 18971}, {67, 15182}, {94, 5169}, {95, 14495}, {114, 1916}, {119, 13109}, {140, 7846}, {147, 3818}, {182, 3329}, {183, 1351}, {187, 10788}, {194, 2996}, {226, 982}, {230, 7607}, {256, 10853}, {264, 2967}, {291, 10854}, {321, 3705}, {338, 18575}, {355, 7976}, {371, 10850}, {372, 10849}, {376, 18842}, {381, 671}, {384, 9737}, {385, 576}, {402, 22698}, {427, 2052}, {459, 3168}, {485, 3102}, {486, 3103}, {493, 22709}, {494, 22710}, {524, 11167}, {538, 3545}, {542, 10033}, {546, 32448}, {574, 11170}, {631, 5188}, {698, 9764}, {726, 3817}, {730, 5587}, {732, 9765}, {736, 7775}, {801, 5020}, {842, 1316}, {858, 16311}, {946, 12782}, {1003, 21166}, {1007, 18906}, {1131, 7000}, {1132, 7374}, {1151, 10841}, {1152, 10842}, {1180, 14773}, {1340, 6039}, {1341, 6040}, {1346, 2592}, {1347, 2593}, {1350, 10007}, {1352, 7774}, {1446, 7185}, {1503, 9300}, {1506, 3399}, {1513, 3094}, {1587, 14244}, {1588, 14229}, {1656, 7868}, {1670, 1676}, {1671, 1677}, {1699, 3097}, {1751, 7413}, {1975, 10983}, {1992, 11172}, {1995, 2986}, {2021, 7737}, {2080, 7771}, {2394, 3906}, {2452, 9139}, {2552, 13580}, {2553, 13581}, {2794, 7753}, {3054, 11668}, {3055, 11669}, {3068, 22720}, {3069, 22721}, {3070, 14238}, {3071, 14234}, {3090, 3934}, {3146, 18845}, {3148, 3425}, {3314, 24206}, {3398, 7878}, {3402, 6210}, {3424, 6776}, {3529, 18843}, {3557, 6178}, {3558, 6177}, {3574, 9290}, {3618, 13354}, {3627, 32516}, {3628, 32521}, {3845, 17503}, {3851, 13108}, {3855, 32450}, {4049, 28565}, {4518, 31395}, {5013, 8719}, {5024, 14485}, {5052, 7612}, {5055, 10302}, {5056, 31276}, {5066, 11055}, {5067, 31239}, {5068, 20081}, {5071, 9466}, {5072, 32520}, {5093, 14614}, {5094, 16080}, {5097, 7766}, {5102, 8667}, {5133, 5392}, {5145, 13478}, {5171, 7824}, {5309, 9302}, {5355, 11623}, {5466, 5996}, {5490, 33344}, {5491, 33345}, {5503, 5969}, {5597, 22668}, {5598, 22672}, {5603, 14839}, {5611, 5981}, {5615, 5980}, {5965, 7837}, {5976, 8781}, {5987, 19140}, {6036, 7806}, {6055, 8587}, {6114, 7684}, {6115, 7685}, {6179, 10104}, {6234, 8842}, {6249, 32476}, {6272, 10515}, {6273, 10514}, {6309, 7764}, {6417, 10845}, {6418, 10846}, {6419, 10847}, {6420, 10848}, {6421, 10837}, {6422, 10838}, {6425, 10843}, {6426, 10844}, {6504, 6997}, {6568, 13785}, {6569, 13665}, {6625, 7385}, {7378, 8796}, {7394, 13579}, {7395, 9917}, {7507, 12143}, {7533, 13582}, {7608, 31489}, {7703, 31127}, {7741, 10079}, {7759, 31982}, {7778, 24256}, {7779, 34507}, {7782, 18502}, {7787, 13335}, {7790, 15980}, {7792, 18583}, {7804, 18860}, {7807, 10256}, {7808, 30270}, {7823, 32152}, {7840, 11178}, {7857, 20576}, {7862, 18806}, {7875, 25555}, {7921, 9863}, {7926, 9996}, {7941, 9983}, {7951, 10063}, {7989, 9902}, {8227, 12263}, {8289, 32135}, {8356, 34733}, {8722, 15482}, {8926, 17795}, {9159, 10989}, {9180, 19912}, {9307, 9747}, {9605, 32467}, {9734, 13586}, {9773, 9880}, {9781, 27375}, {10348, 26316}, {10722, 15484}, {10783, 14243}, {10784, 14228}, {10895, 18982}, {10896, 13077}, {11059, 34087}, {11156, 14036}, {11159, 12117}, {11361, 23698}, {11477, 15271}, {11602, 16809}, {11603, 16808}, {11842, 12042}, {12102, 32523}, {12150, 34473}, {12233, 13380}, {13110, 26470}, {13325, 14633}, {13326, 14632}, {13576, 17756}, {13638, 22722}, {13758, 22723}, {14223, 23350}, {14251, 14265}, {14269, 33698}, {14355, 32716}, {14534, 19544}, {17749, 21554}, {18844, 33703}, {20021, 30499}, {23235, 31859}

▸ Functional Relations34 relations

• X₂₆₂ = midpoint of X_i and X_j for these {i,j}: {1, 22650}, {3, 22728}, {4, 7709}, {39, 22682}, {381, 32447}, {1699, 3097}, {1916, 9772}, {3095, 7697}, {3106, 22694}, {3107, 22693}, {22699, 22700}, {33434, 33435}

• X₂₆₂ = reflection of X_i in X_j for these {i,j}: {1, 22475}, {4, 22682}, {76, 7697}, {376, 21163}, {6194, 15819}, {7697, 5}, {7709, 39}, {7757, 32447}, {9772, 114}, {11257, 7709}, {22676, 3}, {22677, 11261}, {22684, 33463}, {22686, 33462}, {22697, 10}, {22698, 402}, {22712, 2}, {22713, 1}, {22714, 33478}, {22715, 33479}, {22728, 14881}, {22731, 22729}, {22732, 22730}, {33706, 22712}

• X₂₆₂ = isogonal conjugate of X₁₈₂

• X₂₆₂ = isotomic conjugate of X₁₈₃

• X₂₆₂ = complement of X₆₁₉₄

• X₂₆₂ = anticomplement of X₁₅₈₁₉

• X₂₆₂ = isogonal conjugate of the anticomplement of X₂₄₂₀₆

• X₂₆₂ = isogonal conjugate of the complement of X₁₃₅₂

• X₂₆₂ = isotomic conjugate of the anticomplement of X₃₈₁₅

• X₂₆₂ = isotomic conjugate of the complement of X₇₇₇₄

• X₂₆₂ = isotomic conjugate of the isogonal conjugate of X₂₆₃

• X₂₆₂ = isogonal conjugate of the isotomic conjugate of X₃₂₇

• X₂₆₂ = polar conjugate of X₄₅₈

• X₂₆₂ = psi-transform of X₂₂₇₃₅

• X₂₆₂ = antigonal conjugate of isotomic conjugate of X₃₉₀₉₉

• X₂₆₂ = perspector of ABC and Artzt triangle

• X₂₆₂ = X_i-cross conjugate of X_j for these (i,j): {1513, 98}, {3094, 76}, {3815, 2}, {5480, 4}, {9993, 14458}

• X₂₆₂ = X_i-isoconjugate of X_j for these (i,j): {1, 182}, {31, 183}, {32, 3403}, {48, 458}, {63, 10311}, {75, 34396}, {82, 14096}, {163, 23878}, {255, 33971}, {560, 20023}, {662, 3288}, {1933, 8842}, {6784, 24041}

• X₂₆₂ = cevapoint of X_i and X_j for these (i,j): {2, 7774}, {3, 19139}, {6, 3148}, {11, 1491}, {1689, 1690}, {2525, 12037}, {3124, 17415}

• X₂₆₂ = crosssum of X₃₂₈₈ and X₆₇₈₄

• X₂₆₂ = radical center of (Brocard circle reflected in BC, CA, and AB)

• X₂₆₂ = pole wrt polar circle of trilinear polar of X₄₅₈

• X₂₆₂ = X₄₈-isoconjugate (polar conjugate) of X₄₅₈

• X₂₆₂ = trilinear pole of line X₅₂₃X₃₅₆₉

• X₂₆₂ = pole of Lemoine axis wrt orthoptic circle of the Steiner inellipse (a.k.a. {circumcircle, nine-point circle}-inverter)

• X₂₆₂ = perspector of orthoptic circle of the Steiner inellipse (a.k.a. {circumcircle, nine-point circle}-inverter)

• X₂₆₂ = perspector of ABC and 2nd Neuberg triangle

• X₂₆₂ = trilinear product of vertices of 2nd Neuberg triangle

• X₂₆₂ = centroid of X₄PU(1)

• X₂₆₂ = Cundy-Parry Phi transform of X₈₃

• X₂₆₂ = Cundy-Parry Psi transform of X₃₉

• X₂₆₂ = barycentric product X_i*X_j for these {i,j}: {6, 327}, {75, 2186}, {76, 263}, {561, 3402}, {850, 26714}, {2799, 6037}

• X₂₆₂ = barycentric quotient X_i/X_j for these {i,j}: {2, 183}, {4, 458}, {6, 182}, {25, 10311}, {32, 34396}, {39, 14096}, {75, 3403}, {76, 20023}, {141, 14994}, {263, 6}, {327, 76}, {393, 33971}, {512, 3288}, {523, 23878}, {1916, 8842}, {2186, 1}, {2491, 9420}, {3124, 6784}, {3402, 31}, {3815, 15819}, {6037, 2966}, {7735, 9755}, {26714, 110}, {32716, 2715}

• X₂₆₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 6194, 15819}, {2, 9748, 9752}, {2, 9752, 9754}, {2, 14853, 9753}, {2, 22503, 22677}, {3, 10796, 3972}, {3, 11272, 7786}, {4, 39, 11257}, {4, 7736, 9744}, {5, 3095, 76}, {6, 9756, 9755}, {6, 13860, 98}, {39, 31701, 22708}, {39, 31702, 22707}, {114, 19130, 13862}, {194, 3091, 6248}, {381, 11163, 6054}, {485, 3102, 19090}, {486, 3103, 19089}, {1513, 5480, 9993}, {2544, 2545, 7736}, {3090, 12251, 3934}, {3329, 5999, 182}, {3815, 5480, 1513}, {5188, 6683, 631}, {6194, 15819, 22712}, {7777, 13862, 114}, {9737, 10358, 384}, {9748, 9752, 9753}, {9752, 14853, 9748}, {9753, 9754, 9752}, {9755, 9756, 98}, {9755, 13860, 9756}, {10839, 10840, 3}, {11272, 14881, 3}, {19063, 19064, 6}, {21445, 22521, 32}, {22475, 22650, 22713}, {22726, 22727, 31958}, {22729, 22730, 1}, {22731, 22732, 22713}, {31701, 31702, 5475}

Notes

Let H_aH_bH_c be the orthic triangle. X₂₆₂ is the radical center of the Brocard circles of AH_bH_c, BH_cH_a, CH_aH_b. (Randy Hutson, February 10, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = ω. Define B', C' cyclically. Let H_a be the orthocenter of BA'C, and define H_b, H_c cyclically. The lines AH_a, BH_b, CH_c concur in X₂₆₂. (Randy Hutson, July 20, 2016)

Let A' be the apex of the isosceles triangle BA'C constructed intward on BC such that ∠A'BC = ∠A'CB = ω/2. Define B', C' cyclically. Let O_a be the circumcenter of BA'C, and define O_b, O_c cyclically. The lines AO_a, BO_b, CO_c concur in X₂₆₂. (Randy Hutson, July 20, 2016)