X₂₆₅

REFLECTION OF X₃ IN X₁₂₅

X₂₆₅ is the reflection of X₃ in X₁₂₅.

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

X₂₆₅ (REFLECTION OF X₃ IN X₁₂₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 125}

• Related Centers: X₃ (CIRCUMCENTER), X₁₂₅ (CENTER OF JERABEK HYPERBOLA)

Trilinears: \(\sin 2A \csc 3A : :\)

Barycentrics: \(\sin A \sin 2A \csc 3A : :\)

Tripolars: \(a^{2}((a^{2} - b^{2} - c^{2})^{2} - b^{2} c^{2}) : :\)

Search Value (6, 9, 13): \(0.198180509168\)

▸ Alternate Coordinates4 forms

• Trilinears: \(1/(4 \cos A - \sec A) : :\)

• Trilinears: \(\csc(A + \pi /3) - \csc(A - \pi /3) : :\)

• Trilinears: \((\cos A)/(1 - 4 \cos^{2} A) : :\)

• Barycentrics: \((a^{2} - b^{2} - c^{2})/[(a^{2} - b^{2} - c^{2})^{2} - b^{2}c^{2}] : :\)

▸ Linear Combinations1 combination

• X₂₆₅ = \(3\) X₂ - \(5\) X₁₅₀₈₁, \(3\) X₂ - \(4\) X₂₀₃₀₄, \(3\) X₃ - \(4\) X₆₆₉₉, \(2\) X₃ - \(5\) X₁₅₀₂₇, \(2\) X₃ - \(3\) X₁₅₀₆₁, \(3\) X₃ - \(2\) X₁₆₁₆₃, \(5\) X₃ - \(8\) X₂₀₃₉₇, \(3\) X₄ - X₁₄₆, \(3\) X₄ - \(2\) X₁₅₃₉, \(3\) X₄ + X₁₂₃₁₇, \(3\) X₅ - \(2\) X₁₀₂₇₂, \(4\) X₅ - \(3\) X₁₄₆₄₃, \(2\) X₅ - \(3\) X₁₄₆₄₄, \(4\) X₅ - X₂₃₂₃₆, X₂₀ - \(4\) X₂₀₃₇₉, \(3\) X₅₁ - \(2\) X₁₁₅₅₇, X₆₇ + \(2\) X₃₂₂₇₃, X₇₄ - \(3\) X₉₁₄₀, \(3\) X₇₄ - \(2\) X₁₄₆₇₇, \(2\) X₇₄ - \(3\) X₂₀₁₂₆, \(2\) X₉₈ - \(3\) X₁₄₈₄₉, \(2\) X₉₉ - \(3\) X₁₄₈₅₀, \(3\) X₁₁₀ - \(4\) X₁₀₂₇₂, X₁₁₀ - \(4\) X₁₁₈₀₁, \(2\) X₁₁₀ - \(3\) X₁₄₆₄₃, X₁₁₀ - \(3\) X₁₄₆₄₄, \(2\) X₁₁₃ - \(3\) X₃₈₁, \(4\) X₁₁₃ - \(3\) X₅₆₅₅, \(3\) X₁₁₃ - \(2\) X₆₀₅₃, \(3\) X₁₂₅ - \(2\) X₆₆₉₉, \(4\) X₁₂₅ - X₁₂₁₂₁, \(2\) X₁₂₅ + X₁₂₉₀₂, \(4\) X₁₂₅ - \(5\) X₁₅₀₂₇, \(4\) X₁₂₅ - \(3\) X₁₅₀₆₁, \(3\) X₁₂₅ - X₁₆₁₆₃, \(5\) X₁₂₅ - \(4\) X₂₀₃₉₇, \(4\) X₁₄₀ - \(3\) X₁₅₀₃₅, \(4\) X₁₄₀ - \(5\) X₁₅₀₅₉, X₁₄₆ + \(3\) X₃₄₄₈, \(2\) X₁₄₆ - \(3\) X₇₇₂₈, X₁₄₆ - \(6\) X₁₀₁₁₃, \(3\) X₃₈₁ - X₃₉₉, \(9\) X₃₈₁ - \(4\) X₆₀₅₃, \(3\) X₃₈₁ - \(4\) X₇₆₈₇, X₃₈₂ + \(2\) X₁₆₀₀₃, \(2\) X₃₉₉ - \(3\) X₅₆₅₅, \(3\) X₃₉₉ - \(4\) X₆₀₅₃, X₃₉₉ - \(4\) X₇₆₈₇, X₄₇₆ - \(3\) X₅₆₂₇, \(2\) X₄₇₆ - \(3\) X₁₄₉₉₃, \(2\) X₄₇₇ - \(3\) X₁₄₈₅₁

X₂₆₅ lies on the Jerabek circumhyperbola, the Johnson circumconic (see K714), the curves K025, K060, K112, K255, K275, K300, K301, K339, K427, K449, K464, K481, K494, K497, K513, K530, K595, K596, K597, K611, K638, K668, K669, K724, K885, K929, K930, K942, Q106, Q110, Q114, Q125, and also.

▸ Incident Central Lines321 lines

{2, 1511}, {3, 125}, {4, 94}, {5, 49}, {6, 13}, {10, 12778}, {11, 10091}, {12, 10088}, {20, 11270}, {25, 12140}, {30, 74}, {32, 12201}, {51, 11557}, {52, 6145}, {55, 12334}, {56, 18968}, {64, 382}, {65, 79}, {66, 2781}, {67, 511}, {68, 7723}, {69, 328}, {70, 6243}, {71, 12661}, {73, 1807}, {98, 14849}, {99, 14850}, {111, 14846}, {140, 15035}, {148, 18331}, {155, 15317}, {156, 3047}, {182, 19151}, {184, 10254}, {185, 3521}, {186, 30522}, {195, 10112}, {248, 10317}, {290, 316}, {300, 621}, {301, 622}, {355, 6757}, {376, 20421}, {389, 11562}, {402, 12790}, {403, 10540}, {427, 15472}, {477, 14851}, {493, 12894}, {494, 12895}, {515, 13605}, {517, 10693}, {518, 10100}, {520, 14220}, {523, 15453}, {524, 5505}, {526, 10412}, {539, 1568}, {541, 3426}, {546, 1173}, {547, 13392}, {548, 15057}, {549, 15051}, {550, 15055}, {576, 18432}, {590, 10819}, {611, 32297}, {613, 32243}, {615, 10820}, {631, 20396}, {632, 15020}, {690, 6321}, {759, 13743}, {879, 9517}, {895, 3564}, {952, 6742}, {974, 1899}, {1117, 13582}, {1147, 10255}, {1154, 3153}, {1175, 6841}, {1176, 5622}, {1177, 1503}, {1204, 18565}, {1312, 15461}, {1313, 15460}, {1351, 23049}, {1352, 2854}, {1353, 32234}, {1411, 7073}, {1469, 32308}, {1478, 3028}, {1479, 3024}, {1531, 11692}, {1594, 12370}, {1614, 13406}, {1656, 5972}, {1657, 3532}, {1658, 12289}, {1853, 2935}, {1885, 18433}, {1994, 2914}, {2070, 13289}, {2072, 5504}, {2574, 10750}, {2575, 10751}, {2772, 10741}, {2773, 10747}, {2774, 10739}, {2775, 15521}, {2776, 15522}, {2779, 10740}, {2780, 22338}, {2782, 9513}, {2836, 10743}, {2842, 10744}, {2850, 10746}, {2888, 11591}, {2930, 10516}, {2937, 11750}, {2948, 5587}, {3043, 7577}, {3056, 32307}, {3060, 7731}, {3068, 13915}, {3069, 13979}, {3090, 15088}, {3091, 5609}, {3146, 12244}, {3154, 14934}, {3410, 15060}, {3519, 5562}, {3520, 13561}, {3526, 6723}, {3527, 3843}, {3530, 15036}, {3531, 14269}, {3541, 15114}, {3543, 11738}, {3545, 9143}, {3548, 18466}, {3549, 18945}, {3567, 15102}, {3574, 11536}, {3583, 7727}, {3589, 15462}, {3627, 10721}, {3628, 15025}, {3629, 15093}, {3657, 8674}, {3839, 14491}, {3845, 10706}, {3851, 16534}, {5050, 15118}, {5055, 5642}, {5072, 15046}, {5073, 10990}, {5076, 22334}, {5079, 15039}, {5093, 5095}, {5097, 15432}, {5102, 16176}, {5446, 11572}, {5447, 26861}, {5448, 15002}, {5462, 16223}, {5480, 9970}, {5576, 12241}, {5597, 12466}, {5598, 12467}, {5621, 12083}, {5648, 11178}, {5886, 11720}, {5889, 12281}, {5890, 12270}, {5891, 13622}, {5900, 13391}, {5944, 12254}, {5946, 11561}, {5987, 13862}, {6000, 10688}, {6146, 10024}, {6214, 7733}, {6215, 7732}, {6221, 8994}, {6284, 10065}, {6334, 9033}, {6398, 13969}, {6413, 12891}, {6414, 12892}, {6528, 8795}, {6593, 14561}, {6639, 19467}, {6640, 12118}, {6643, 13416}, {6776, 25320}, {6798, 31392}, {7354, 10081}, {7387, 13171}, {7422, 31127}, {7426, 15362}, {7507, 19504}, {7512, 13470}, {7517, 10117}, {7528, 15465}, {7547, 12161}, {7579, 32235}, {7583, 19111}, {7584, 19110}, {7689, 18562}, {7693, 13364}, {7699, 11422}, {7978, 22791}, {8044, 14616}, {8200, 13208}, {8207, 13209}, {8220, 13215}, {8221, 13216}, {8571, 14585}, {8612, 18416}, {8673, 18557}, {8976, 8998}, {9818, 12168}, {9826, 12099}, {9833, 15647}, {9919, 18534}, {9955, 11699}, {9996, 13210}, {10095, 22804}, {10097, 14582}, {10152, 18507}, {10201, 20773}, {10246, 11735}, {10263, 15101}, {10293, 14915}, {10575, 17855}, {10663, 18468}, {10664, 18470}, {10752, 21850}, {10767, 22938}, {10796, 13193}, {10942, 13217}, {10943, 13218}, {11061, 14853}, {11138, 11582}, {11139, 11581}, {11250, 23294}, {11251, 15395}, {11262, 32369}, {11411, 15077}, {11442, 12825}, {11449, 11704}, {11457, 17854}, {11459, 12273}, {11499, 13204}, {11563, 14157}, {11564, 18572}, {11585, 22808}, {11694, 15699}, {11709, 18481}, {11723, 18493}, {11746, 14542}, {11807, 15321}, {11818, 12824}, {11898, 32275}, {12038, 16665}, {12111, 12284}, {12162, 22466}, {12165, 18386}, {12192, 14880}, {12278, 26917}, {12359, 18442}, {12429, 15316}, {12584, 24206}, {12773, 26700}, {12811, 15029}, {12828, 15473}, {12888, 18455}, {13160, 13353}, {13201, 17711}, {13291, 14695}, {13292, 23047}, {13293, 20299}, {13340, 14791}, {13352, 15131}, {13403, 14130}, {13598, 18555}, {13603, 15687}, {13951, 13990}, {14128, 18368}, {14216, 31725}, {14452, 19658}, {14487, 14893}, {14639, 15342}, {14708, 18912}, {14859, 19552}, {14919, 20123}, {14940, 32171}, {15021, 15704}, {15042, 15693}, {15136, 18441}, {15357, 23698}, {15538, 15550}, {15740, 16270}, {16010, 25330}, {16168, 17511}, {17040, 18537}, {18281, 25487}, {18376, 18434}, {18414, 19484}, {18415, 19485}, {18462, 19482}, {18463, 19483}, {18489, 32255}, {18570, 23293}, {18909, 31371}, {19402, 32353}, {22586, 22758}, {23329, 25564}, {25335, 32271}, {29012, 32305}

▸ Functional Relations35 relations

• X₂₆₅ = midpoint of X_i and X_j for these {i,j}: {3, 12902}, {4, 3448}, {74, 10733}, {146, 12317}, {148, 18331}, {382, 10620}, {1351, 32306}, {3146, 12244}, {5889, 12281}, {6321, 15545}, {7731, 15100}, {10263, 15101}, {10721, 15054}, {12111, 12284}, {12293, 12302}, {12295, 16003}, {12803, 12804}, {13213, 13214}, {14508, 14989}, {21649, 21650}

• X₂₆₅ = reflection of X_i in X_j for these {i,j}: {3, 125}, {4, 10113}, {5, 11801}, {20, 12041}, {52, 11800}, {54, 11804}, {74, 10264}, {98, 15535}, {110, 5}, {113, 7687}, {146, 1539}, {182, 20301}, {185, 11806}, {382, 12295}, {399, 113}, {477, 16340}, {1352, 32274}, {1511, 20304}, {1657, 16111}, {1986, 12236}, {3581, 3580}, {5504, 23306}, {5648, 11178}, {5655, 381}, {6102, 13358}, {7722, 6102}, {7723, 15738}, {7728, 4}, {7978, 22791}, {9833, 15647}, {9970, 5480}, {10114, 19481}, {10540, 403}, {10575, 17855}, {10620, 16003}, {10706, 3845}, {10721, 3627}, {10752, 21850}, {10767, 22938}, {11562, 389}, {11579, 25328}, {11699, 9955}, {11898, 32275}, {12041, 20379}, {12121, 3}, {12308, 15063}, {12368, 18480}, {12383, 1511}, {12584, 24206}, {12778, 10}, {12790, 402}, {12893, 5449}, {13293, 20299}, {13417, 5446}, {14157, 11563}, {14559, 14356}, {14643, 14644}, {14683, 5609}, {14934, 3154}, {14982, 3818}, {14989, 21269}, {14993, 5627}, {15131, 23325}, {16111, 20417}, {16163, 6699}, {18332, 115}, {18403, 13851}, {18436, 7723}, {18439, 12292}, {18481, 11709}, {19140, 19130}, {19506, 18383}, {20126, 9140}, {20127, 74}, {22115, 2072}, {22584, 21650}, {23236, 110}, {24981, 16534}, {25711, 11746}, {30714, 5972}, {32233, 182}, {32234, 1353}, {32609, 23515}

• X₂₆₅ = isogonal conjugate of X₁₈₆

• X₂₆₅ = isotomic conjugate of X₃₄₀

• X₂₆₅ = complement of X₁₂₃₈₃

• X₂₆₅ = anticomplement of X₁₅₁₁

• X₂₆₅ = circumcircle-inverse of X₅₉₆₁

• X₂₆₅ = nine-point-circle-inverse of X₁₅₃₆₇

• X₂₆₅ = polar-circle-inverse of X₁₉₈₆

• X₂₆₅ = circumcircle-of-anticomplementary-triangle-inverse of X₁₂₃₁₇

• X₂₆₅ = polar conjugate of X₁₄₁₆₅

• X₂₆₅ = antigonal image of X₃

• X₂₆₅ = syngonal conjugate of X₅

• X₂₆₅ = X_i-anticomplementary conjugate of X_j for these (i,j): {2159, 18301}, {2166, 146}, {2349, 1272}, {5627, 8}, {11079, 6360}, {15395, 6758}

• X₂₆₅ = X_i-Ceva conjugate of X_j for these (i,j): {94, 1989}, {1141, 5961}, {12028, 3}, {14254, 14993}, {18817, 18883}

• X₂₆₅ = X_i-cross conjugate of X_j for these (i,j): {3, 15392}, {1531, 4846}, {3284, 2}, {13754, 3}, {13851, 4}, {14391, 648}, {18403, 3521}, {21649, 5504}, {21650, 74}

• X₂₆₅ = cevapoint of X_i and X_j for these (i,j): {4, 6761}, {5, 30}, {125, 9033}, {184, 9380}, {520, 1650}, {621, 622}, {686, 20975}

• X₂₆₅ = crosspoint of X_i and X_j for these (i,j): {94, 328}, {1494, 2986}

• X₂₆₅ = crosssum of X_i and X_j for these (i,j): {3, 2931}, {526, 16186}, {1495, 3003}, {2088, 14270}

• X₂₆₅ = trilinear pole of line {216, 647}

• X₂₆₅ = crossdifference of every pair of points on line {526, 2081}

• X₂₆₅ = X₈-beth conjugate of X₁₂₇₇₈

• X₂₆₅ = X_i-zayin conjugate of X_j for these (i,j): {1, 186}, {1021, 2624}

• X₂₆₅ = X₁₀₄₁₂-line conjugate of X₅₂₆

• X₂₆₅ = X_i-vertex conjugate of X_j for these (i,j): {3, 11744}, {186, 6344}, {1177, 5504}

• X₂₆₅ = X_i-isoconjugate of X_j for these (i,j): {1, 186}, {4, 6149}, {19, 323}, {31, 340}, {35, 1870}, {36, 6198}, {47, 5962}, {48, 14165}, {50, 92}, {112, 32679}, {158, 22115}, {162, 526}, {240, 14355}, {275, 2290}, {320, 14975}, {470, 2152}, {471, 2151}, {562, 2964}, {648, 2624}, {661, 14590}, {811, 14270}, {860, 17104}, {1154, 2190}, {1399, 5081}, {1464, 11107}, {1577, 14591}, {1784, 14385}, {1969, 19627}, {1973, 7799}, {2148, 14918}, {2159, 14920}, {2166, 3043}, {2167, 11062}, {2174, 17923}, {2361, 7282}, {2594, 17515}, {2605, 4242}, {3268, 32676}, {8552, 24019}, {16186, 24000}

• X₂₆₅ = barycentric product X_i*X_j for these {i,j}: {3, 94}, {6, 328}, {63, 2166}, {68, 18883}, {69, 1989}, {99, 14582}, {110, 14592}, {184, 20573}, {287, 14356}, {305, 11060}, {311, 11077}, {343, 1141}, {394, 6344}, {476, 525}, {577, 18817}, {656, 32680}, {850, 32662}, {1304, 18557}, {1807, 30690}, {2410, 14220}, {3260, 11079}, {3267, 14560}, {3519, 30529}, {3580, 12028}, {3926, 18384}, {4558, 10412}, {4563, 15475}, {5392, 5961}, {5627, 11064}, {7100, 18359}, {7799, 14595}, {10217, 11092}, {10218, 11078}, {14208, 32678}, {14254, 14919}, {14559, 14977}, {16077, 18558}

• X₂₆₅ = barycentric quotient X_i/X_j for these {i,j}: {2, 340}, {3, 323}, {4, 14165}, {5, 14918}, {6, 186}, {13, 471}, {14, 470}, {30, 14920}, {48, 6149}, {50, 3043}, {51, 11062}, {69, 7799}, {79, 17923}, {94, 264}, {110, 14590}, {184, 50}, {216, 1154}, {248, 14355}, {328, 76}, {343, 1273}, {476, 648}, {520, 8552}, {525, 3268}, {577, 22115}, {647, 526}, {656, 32679}, {810, 2624}, {1141, 275}, {1576, 14591}, {1807, 3219}, {1989, 4}, {2006, 7282}, {2160, 1870}, {2161, 6198}, {2165, 5962}, {2166, 92}, {2341, 11107}, {2437, 7480}, {2963, 562}, {3003, 1986}, {3049, 14270}, {3269, 16186}, {3284, 1511}, {3457, 8740}, {3458, 8739}, {4558, 10411}, {5158, 3581}, {5627, 16080}, {5961, 1993}, {6344, 2052}, {7100, 3218}, {7110, 5081}, {8014, 23714}, {8015, 23715}, {8606, 2323}, {8818, 860}, {9033, 5664}, {9380, 11597}, {10097, 9213}, {10217, 11078}, {10218, 11092}, {10412, 14618}, {11060, 25}, {11063, 2914}, {11064, 6148}, {11077, 54}, {11079, 74}, {11083, 10632}, {11088, 10633}, {12028, 2986}, {14220, 2411}, {14356, 297}, {14559, 4235}, {14560, 112}, {14575, 19627}, {14582, 523}, {14583, 1990}, {14592, 850}, {14595, 1989}, {15392, 13582}, {15451, 2081}, {15475, 2501}, {18384, 393}, {18479, 5158}, {18558, 9033}, {18817, 18027}, {18877, 14385}, {18883, 317}, {20573, 18022}, {20975, 2088}, {23968, 7473}, {30529, 32002}, {32662, 110}, {32678, 162}, {32680, 811}

• X₂₆₅ = Johnson-circumconic antipode of X₁₁₀

• X₂₆₅ = perspector of ABC and 2nd isogonal triangle of X₄

• X₂₆₅ = perspector of ABC and Ehrmann side-triangle

• X₂₆₅ = perspector, wrt Ehrmann side-triangle, of Ehrmann conic

• X₂₆₅ = pole of line X₃X₅₂₃ (the line of the Ehrmann cross-triangle) wrt the Ehrmann conic

• X₂₆₅ = homothetic center of orthic triangle and cross-triangle of Ehrmann side- and Ehrmann vertex-triangles

• X₂₆₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 12383, 1511}, {2, 15081, 20304}, {3, 125, 15061}, {4, 146, 1539}, {4, 12317, 146}, {4, 18300, 18576}, {4, 32140, 18439}, {5, 110, 14643}, {5, 11801, 14644}, {5, 12022, 567}, {5, 14516, 18350}, {13, 14, 1989}, {52, 18383, 31724}, {52, 31724, 15800}, {68, 18404, 18436}, {74, 9140, 10264}, {74, 10264, 20126}, {79, 80, 2166}, {94, 18300, 4}, {98, 15535, 14849}, {110, 12228, 11597}, {110, 14644, 5}, {113, 399, 5655}, {113, 7687, 381}, {113, 10111, 18445}, {113, 19481, 15087}, {125, 12902, 12121}, {125, 16163, 6699}, {125, 21659, 32607}, {146, 1539, 7728}, {146, 3448, 12317}, {381, 399, 113}, {381, 15087, 18388}, {399, 19456, 18445}, {477, 16340, 14851}, {567, 11597, 12228}, {568, 18430, 4}, {1511, 20304, 2}, {1656, 32609, 5972}, {1657, 15041, 16111}, {1986, 12236, 568}, {2931, 19457, 3}, {3060, 15100, 7731}, {3448, 10113, 7728}, {3583, 7727, 12374}, {3585, 19470, 12373}, {5449, 21659, 3}, {5622, 19138, 19129}, {5889, 18394, 18377}, {5972, 23515, 1656}, {5972, 30714, 32609}, {6102, 18379, 4}, {6344, 23956, 4}, {6564, 6565, 9220}, {6699, 16163, 3}, {7687, 10114, 18388}, {8836, 8838, 18883}, {9140, 10733, 74}, {10114, 18388, 12227}, {10264, 10733, 20127}, {11746, 25711, 16222}, {12121, 15027, 15061}, {12121, 15061, 3}, {12317, 18932, 18917}, {12319, 18933, 18531}, {12383, 15081, 2}, {12893, 32607, 3}, {13851, 21649, 19479}, {14094, 15044, 546}, {14643, 23236, 110}, {14852, 18396, 3}, {15025, 15034, 3628}, {15027, 15061, 125}, {15035, 15059, 140}, {16111, 20417, 15041}, {16770, 16771, 94}, {17838, 18451, 399}, {18390, 18474, 381}, {19051, 19052, 6}, {20126, 20127, 74}, {23515, 30714, 5972}.

Notes

Let P = X₇₄, H = X₄, H' =H-of-BCP, H″ = H-of-CAP, and H’’’ = H-of ABP. Then X₂₆₅ is the circumcenter of the cyclic quadrilateral HH’H’’H’’’. (Randy Hutson, 9/23/2011)

Let A' be the reflection in line BC of the A-vertex of the tangential triangle, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur in X₂₆₅. Let A″ be the reflection in line BC of the A-vertex of the anticevian triangle of X₅, and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X₂₆₅. (Randy Hutson, August 26, 2014)

Let A*B*C* be the Kosnita triangle. Let A' be the orthopole of line B*C*, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₆₅. (Randy Hutson, August 26, 2014)

Let A'B'C' be the X₃-Fuhrmann triangle. Let A″ be the reflection of A in line B'C', and define B″ and C″ cyclically. Then A″B″C″ is inversely similar to ABC, with similtude center X₂₆₅, and A″B″C″ is perspective to ABC with persepctor X₇₄. (Randy Hutson, August 26, 2014)

Let Ref_aRef_bRef_c be the reflection triangle. Let L be the line through Ref_a parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L',M', N' concur in X₂₆₅. (Randy Hutson, August 26, 2014)

Let Ref_aRef_bRef_c be the reflection triangle. Let A″ be the trilinear pole of line Ref_bRef_c, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₅. Let A* be the trilinear pole, wrt Ref_aRef_bRef_c, of line B″C″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X₂₆₅. (Randy Hutson, July 20, 2016)

Let A' be the point such that triangle A'BC is directly similar to the orthic triangle, and define B', C' cyclically. The lines AA', BB', CC' concur in X₂₆₅. If ‘inversely’ is substituted for ‘directly’, the lines concur in X₃. (Randy Hutson, July 20, 2016)

Let A' be the isogonal conjugate of A wrt the A-altimedial triangle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₆₅. (Randy Hutson, November 2, 2017)

Let L_a be the line through A parallel to the Euler line of the A-altimedial triangle, and define L_b and L_c cyclically. Lines L_a, L_b, L_c concur in X₂₆₅. (Randy Hutson, November 2, 2017)

Let AA₁A₂, BB₁B₂, CC₁C₂ be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the crosssum of A₁ and A₂, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₆₅. (Randy Hutson, October 8, 2019)

Let O_A be the circle centered at the A-vertex of the orthocentroidal triangle and passing through A; define O_B and O_C cyclically. X₂₆₅ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)