X₃₀₆₈

INTERSECTION OF X₂X_6 AND X₄X_371

X₃₀₆₈ = complement of X₁₂₇₀

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

X₃₀₆₈ (INTERSECTION OF X₂X_6 AND X₄X_371) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 1270}

• Related Centers: X₂ (CENTROID), X₁₂₇₀ (ISOTOMIC CONJUGATE OF X₁₁₃₁)

Trilinears: \(\sin A + \sin B \sin C : :\)

Barycentrics: \((\sin A)(\sin A + \sin B \sin C) : :\)

Search Value (6, 9, 13): \(1.535810268985\)

▸ Alternate Coordinates4 forms

• Trilinears: \((S + a^{2})/a : :\)

• Trilinears: \(\cos A + \sin A + \cos B \cos C : :\)

• Barycentrics: \(a^{2} + S : :\)

• Barycentrics: \(S_{B} + S_{C} + S : :\)

▸ Incident Central Lines341 lines

{1, 1336}, {2, 6}, {3, 1587}, {4, 371}, {5, 1588}, {8, 7969}, {9, 5393}, {10, 13893}, {11, 13895}, {12, 13896}, {13, 13917}, {14, 13916}, {17, 22921}, {18, 22876}, {19, 1659}, {20, 1151}, {24, 8276}, {25, 13884}, {30, 6221}, {32, 638}, {39, 8963}, {40, 13912}, {53, 19410}, {54, 8995}, {55, 13887}, {56, 18965}, {64, 8991}, {68, 13909}, {74, 8994}, {76, 8992}, {79, 16148}, {80, 8988}, {83, 5491}, {84, 8987}, {98, 8980}, {99, 8997}, {100, 13922}, {104, 13913}, {110, 8998}, {112, 13923}, {113, 19060}, {114, 19056}, {115, 19109}, {119, 19082}, {125, 19111}, {127, 19115}, {132, 19094}, {140, 3312}, {148, 13657}, {171, 606}, {184, 18924}, {186, 9682}, {214, 19078}, {216, 1590}, {238, 605}, {239, 1267}, {262, 22720}, {265, 13915}, {372, 631}, {376, 6200}, {381, 6199}, {382, 23253}, {387, 2047}, {388, 2067}, {393, 493}, {402, 13894}, {427, 5410}, {468, 5411}, {486, 3090}, {487, 5286}, {488, 6423}, {489, 5254}, {490, 3053}, {494, 13900}, {497, 2066}, {498, 3301}, {499, 3299}, {511, 22717}, {515, 9583}, {516, 9616}, {530, 13646}, {531, 13645}, {532, 22918}, {533, 22873}, {538, 13647}, {542, 13640}, {543, 13642}, {546, 23263}, {548, 6455}, {549, 6398}, {550, 6449}, {571, 13439}, {577, 1589}, {588, 2165}, {618, 19074}, {619, 19076}, {620, 19108}, {629, 19070}, {630, 19072}, {632, 6428}, {637, 1504}, {640, 7375}, {641, 13771}, {642, 19105}, {671, 13908}, {754, 13648}, {894, 5391}, {920, 1123}, {946, 1702}, {958, 19014}, {1001, 18999}, {1030, 21566}, {1033, 15210}, {1075, 8954}, {1078, 18993}, {1124, 3086}, {1125, 13959}, {1131, 3146}, {1132, 3590}, {1147, 19062}, {1152, 3523}, {1181, 6807}, {1209, 19096}, {1249, 3535}, {1297, 13918}, {1327, 13920}, {1328, 13848}, {1335, 3085}, {1370, 11417}, {1376, 19000}, {1378, 19843}, {1449, 5405}, {1503, 7374}, {1511, 19052}, {1579, 7400}, {1583, 8573}, {1586, 3087}, {1599, 1609}, {1650, 19018}, {1656, 6417}, {1657, 6407}, {1698, 13936}, {1703, 6684}, {1788, 2362}, {1899, 18923}, {2043, 5335}, {2044, 5334}, {2271, 21909}, {2482, 19058}, {2548, 5058}, {2550, 5415}, {2883, 19088}, {3035, 19112}, {3071, 3091}, {3074, 3077}, {3075, 3076}, {3083, 3554}, {3084, 3553}, {3088, 3093}, {3089, 3092}, {3096, 19012}, {3103, 12251}, {3147, 10881}, {3183, 22838}, {3186, 8956}, {3297, 14986}, {3317, 10195}, {3462, 8955}, {3485, 16232}, {3522, 6409}, {3524, 6396}, {3525, 5420}, {3526, 6418}, {3528, 9680}, {3529, 6453}, {3530, 6450}, {3534, 6445}, {3543, 6437}, {3545, 6565}, {3594, 10303}, {3616, 7968}, {3624, 13971}, {3627, 6447}, {3628, 6427}, {3634, 13947}, {3647, 19080}, {3832, 23261}, {3855, 23275}, {3934, 19089}, {4254, 16433}, {4296, 9634}, {4297, 9615}, {4423, 13940}, {5020, 19005}, {5021, 21992}, {5054, 6395}, {5055, 18510}, {5056, 6431}, {5059, 6429}, {5067, 10577}, {5070, 6500}, {5120, 16432}, {5124, 21567}, {5200, 10132}, {5218, 5414}, {5305, 11313}, {5409, 6805}, {5413, 6353}, {5432, 19037}, {5433, 18995}, {5449, 19061}, {5461, 19057}, {5480, 7000}, {5552, 19048}, {5597, 13890}, {5598, 13891}, {5599, 19008}, {5600, 19010}, {5889, 12239}, {5972, 19110}, {6036, 19055}, {6118, 19102}, {6119, 19104}, {6193, 8909}, {6202, 8396}, {6260, 19068}, {6292, 19092}, {6302, 9112}, {6303, 9113}, {6392, 6462}, {6410, 15717}, {6411, 10304}, {6412, 15692}, {6424, 7388}, {6438, 15708}, {6446, 15693}, {6448, 12108}, {6451, 8703}, {6452, 12100}, {6456, 15712}, {6468, 15683}, {6480, 11001}, {6481, 15719}, {6501, 13961}, {6502, 7288}, {6519, 15704}, {6643, 10897}, {6669, 19073}, {6670, 19075}, {6673, 19071}, {6674, 19069}, {6689, 19095}, {6696, 19087}, {6699, 19059}, {6701, 19079}, {6702, 19077}, {6704, 19091}, {6705, 19067}, {6713, 19081}, {6720, 19114}, {6776, 6811}, {6808, 10982}, {6813, 14853}, {7160, 13914}, {7386, 11513}, {7392, 10961}, {7493, 11418}, {7494, 11514}, {7615, 13660}, {7737, 9675}, {7738, 11293}, {7815, 13938}, {7846, 19011}, {8222, 19032}, {8223, 19034}, {8280, 8889}, {8383, 19219}, {8882, 16032}, {9648, 15338}, {9663, 15326}, {9681, 22644}, {9683, 12088}, {9686, 13346}, {9690, 15681}, {9691, 17800}, {9694, 11413}, {9695, 12082}, {9722, 15234}, {9732, 21737}, {9780, 13973}, {9862, 13674}, {10192, 17820}, {10198, 13965}, {10200, 13964}, {10266, 13919}, {10527, 19050}, {10533, 11206}, {10665, 11411}, {10819, 12383}, {11265, 14790}, {11284, 13943}, {11294, 12323}, {11442, 11447}, {11457, 11462}, {11485, 18585}, {11486, 15765}, {11542, 18587}, {11543, 18586}, {12221, 13881}, {12240, 15043}, {12317, 12375}, {12318, 12424}, {12319, 12891}, {12320, 12960}, {12321, 12961}, {12322, 12962}, {12324, 12964}, {12325, 12965}, {12376, 20125}, {12864, 19086}, {13025, 13045}, {13026, 13046}, {13089, 19098}, {13203, 13287}, {13650, 13711}, {13662, 13720}, {13701, 22541}, {13774, 15118}, {13821, 19100}, {13832, 13833}, {15183, 19017}, {15722, 17851}, {15819, 19064}, {18457, 18531}, {19051, 20304}, {19084, 22966}, {19174, 19183}, {19420, 19436}, {19421, 19438}, {22466, 22976}, {22555, 22960}

▸ Functional Relations10 relations

• X₃₀₆₈ = isogonal conjugate of X₄₉₃

• X₃₀₆₈ = complement of X₁₂₇₀

• X₃₀₆₈ = X_i-Ceva conjugate of X_j for these (i,j): (393, 3069), (1585, 4)

• X₃₀₆₈ = crosspoint of X₂ and X₁₁₃₁

• X₃₀₆₈ = crosssum of X₆ and X₁₁₅₁

• X₃₀₆₈ = {X₂,X₆}-harmonic conjugate of X₃₀₆₉

• X₃₀₆₈ = X₂-of-1st-tri-squares-triangle

• X₃₀₆₈ = X₂-of-1st-tri-squares-central-triangle

• X₃₀₆₈ = orthologic center of these triangles: 1st tri-squares to outer-Vecten

• X₃₀₆₈ = homothetic center of ABC and 3rd tri-squares central triangle

Notes

X₃₀₆₈: Let A'B'C' be the outer Vecten triangle. Let A″ be the trilinear pole of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₀₆₈. (Randy Hutson, July 21, 2017)