X567

ORTHOGONAL OF X50

X567 = Brocard-circle-inverse of X568
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 3, 6 }: Brocard Axis { 5, 49 }: Central Line L(5, 49) { 184, 381 }: Central Line L(184, 381) { 546, 1614 }: Central Line L(546, 1614) { 1147, 1656 }: Central Line L(1147, 1656) { 1154, 1994 }: Central Line L(1154, 1994) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_546 = MIDPOINT OF X(4) AND X(5) (0.5842, 0.3223) X_1614 = TCC-PERSPECTOR OF X(54) (0.5582, -0.0024) X_1147 = ISOGONAL CONJUGATE OF X(847) (0.8329, 0.3027) X_1656 = INTERSECTION OF EULER LINE AND LINEX(17)X(18) (0.5449, 0.2195) X_1154 = ISOGONAL CONJUGATE OF X(1141) (0.8932, 0.6667) X_1994 = ORTHOCORRESPONDENT OF X(5) (0.6334, 0.3031) X_567 = ORTHOGONAL OF X(50) (0.5767, 0.2287) A B C X3 X6 X5 X49 X184 X381 X546 X1614 X1147 X1656 X1154 X1994 X567 (a, b, c) = (5, 6, 7) | AB = 1

X567 (ORTHOGONAL OF X50) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cos(B - C) + 2 \cos(C - A)\cos(A - B) : :\)
Search Value (6, 9, 13): \(-0.059795309774\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\sin A + \cos A \cot D/2 : :\), where cot \(D/2 = (a^{2} + b^{2} + c^{2} - 6R^{2})/(4 \operatorname{area})\), where \(R = abc/(4 \operatorname{area})\)

X567 lies on the curve Q166.

▸ Incident Central Lines247 lines

{2, 22115}, {3, 6}, {4, 7578}, {5, 49}, {13, 11134}, {14, 11137}, {22, 39522}, {30, 5012}, {51, 2070}, {74, 45956}, {115, 9604}, {125, 43573}, {140, 3580}, {143, 7488}, {156, 3091}, {184, 381}, {185, 10620}, {186, 5946}, {195, 5562}, {206, 23049}, {215, 7951}, {323, 7550}, {373, 32609}, {382, 11424}, {399, 15030}, {498, 9666}, {499, 9653}, {546, 1614}, {547, 40111}, {549, 43574}, {550, 43576}, {597, 15462}, {631, 37644}, {1092, 3526}, {1147, 1656}, {1154, 1994}, {1173, 12107}, {1176, 21850}, {1180, 39524}, {1181, 10938}, {1199, 6102}, {1209, 10112}, {1352, 14787}, {1437, 37251}, {1493, 11591}, {1495, 7545}, {1506, 9603}, {1511, 13363}, {1657, 8717}, {1658, 3567}, {1976, 37345}, {1986, 3520}, {1993, 7514}, {2072, 23292}, {2477, 7741}, {2931, 12038}, {2937, 5446}, {3043, 20304}, {3060, 7502}, {3066, 7506}, {3090, 9545}, {3200, 37835}, {3201, 37832}, {3292, 10170}, {3518, 5944}, {3519, 12899}, {3521, 18560}, {3527, 9714}, {3545, 9544}, {3574, 31724}, {3796, 12083}, {3814, 9702}, {3843, 6759}, {3845, 14157}, {3851, 9704}, {5054, 43650}, {5055, 9306}, {5070, 16187}, {5169, 34514}, {5422, 6644}, {5462, 13367}, {5476, 18374}, {5480, 7540}, {5504, 37648}, {5544, 12309}, {5576, 6146}, {5609, 15052}, {5622, 20126}, {5640, 11464}, {5643, 15034}, {5654, 41615}, {5663, 7527}, {5876, 32136}, {5885, 43610}, {5890, 11454}, {5891, 34986}, {5921, 7404}, {6033, 39834}, {6101, 37126}, {6143, 43816}, {6193, 14786}, {6321, 39805}, {6636, 13391}, {6639, 39571}, {6723, 43817}, {6800, 7530}, {7399, 43595}, {7403, 31804}, {7503, 12161}, {7509, 16266}, {7512, 10263}, {7516, 21766}, {7517, 10982}, {7526, 7592}, {7528, 18925}, {7555, 15107}, {7556, 11002}, {7575, 15019}, {7579, 15139}, {7603, 9696}, {7728, 13198}, {7769, 10411}, {7827, 14355}, {7988, 9621}, {7989, 9622}, {8253, 9676}, {8550, 11579}, {8703, 13482}, {9677, 23261}, {9697, 39565}, {9701, 25639}, {9707, 13861}, {9777, 14070}, {9781, 37440}, {9818, 11402}, {9833, 18382}, {9920, 10282}, {10024, 12241}, {10110, 18378}, {10226, 43600}, {10254, 18390}, {10257, 45298}, {10264, 43578}, {10274, 15432}, {10574, 11250}, {10601, 47391}, {10619, 45286}, {11004, 33533}, {11422, 11459}, {11423, 12111}, {11427, 18531}, {11428, 18453}, {11429, 18455}, {11449, 15024}, {11456, 31861}, {11799, 16657}, {11808, 44515}, {11820, 47527}, {12006, 12236}, {12121, 38323}, {12162, 12308}, {12164, 34801}, {12225, 15800}, {12227, 22584}, {12229, 22809}, {12230, 22810}, {12233, 18563}, {12234, 22815}, {12295, 13403}, {12307, 14531}, {12370, 13160}, {13011, 22813}, {13012, 22814}, {13142, 34002}, {13321, 15004}, {13364, 13595}, {13366, 13754}, {13451, 37936}, {13491, 14865}, {13561, 43808}, {13564, 45186}, {13567, 45967}, {14560, 18121}, {14831, 32608}, {14845, 21308}, {14848, 19136}, {14855, 35452}, {14869, 46865}, {14912, 18917}, {14926, 41597}, {15026, 32171}, {15027, 15132}, {15035, 43584}, {15036, 43597}, {15043, 37814}, {15051, 43804}, {15053, 15646}, {15054, 43596}, {15059, 34331}, {15061, 15463}, {15062, 43602}, {15472, 20127}, {15620, 30536}, {15699, 43572}, {15720, 43652}, {15805, 35602}, {16003, 33749}, {16226, 37955}, {17809, 18451}, {17835, 43807}, {18281, 18911}, {18388, 18403}, {18434, 40276}, {18447, 19365}, {18462, 19408}, {18463, 19409}, {18494, 44077}, {18502, 40643}, {18524, 20986}, {18952, 37119}, {19127, 20423}, {19128, 37458}, {19138, 43815}, {19361, 32322}, {22529, 22808}, {23328, 43810}, {25555, 30714}, {25739, 39504}, {27866, 34153}, {32068, 44673}, {32127, 32284}, {32235, 36253}, {32245, 32272}, {32345, 41725}, {32379, 41362}, {34007, 43818}, {34396, 35934}, {35498, 43604}, {37347, 37649}, {39562, 40673}, {40114, 44275}, {40441, 45089}, {43605, 45959}, {43612, 43806}, {43831, 46686}, {43839, 46085}, {44085, 45923}

▸ Functional Relations12 relations

  • X567 = midpoint of Xi and Xj for these {i,j}: {1994, 35921}, {5012, 15033}, {7527, 15032}

  • X567 = reflection of Xi in Xj for these {i,j}: {3, 37513}, {15087, 13366}, {37347, 37649}, {41171, 5}

  • X567 = isogonal conjugate of X9221

  • X567 = Brocard-circle-inverse of X568

  • X567 = crosssum of X11 and X18116

  • X567 = crossdifference of every pair of points on line {523, 2081}

  • X567 = crosspoint of isogonal conjugates of PU(5)

  • X567 = homothetic center of Ehrmann side-triangle and 1st anti-Conway triangle

  • X567 = X3-Dao conjugate of X9221

  • X567 = X1-isoconjugate of X9221

  • X567 = barycentric quotient X6/X9221

  • X567 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {3, 6, 568}, {3, 182, 13339}, {3, 568, 3581}, {3, 569, 13353}, {3, 578, 37472}, {3, 1351, 37494}, {3, 11426, 36749}, {3, 11432, 37490}, {3, 13352, 37477}, {3, 13353, 37471}, {3, 14627, 52}, {3, 15037, 9730}, {3, 36747, 37484}, {3, 36749, 6243}, {3, 36753, 37481}, {3, 37472, 37495}, {3, 37506, 14805}, {5, 49, 18350}, {5, 54, 49}, {5, 12022, 265}, {5, 44076, 6288}, {6, 14805, 3581}, {6, 37506, 3}, {51, 18475, 2070}, {52, 37505, 14627}, {54, 13434, 5}, {54, 43598, 9706}, {61, 62, 41335}, {143, 10610, 7488}, {182, 578, 13352}, {182, 13339, 37471}, {182, 13352, 3}, {184, 381, 10540}, {186, 34545, 5946}, {195, 34864, 5562}, {265, 12228, 11597}, {323, 7550, 15067}, {371, 372, 566}, {381, 18396, 18430}, {568, 13353, 19129}, {568, 14805, 3}, {568, 18438, 6243}, {569, 578, 3}, {569, 13352, 182}, {569, 37472, 37471}, {575, 9730, 15037}, {575, 11430, 9730}, {578, 13353, 37495}, {1199, 14118, 6102}, {1351, 37494, 6243}, {1993, 7514, 23039}, {2070, 15038, 51}, {3796, 44413, 12083}, {3851, 9704, 10539}, {5055, 9703, 9306}, {5085, 37483, 3}, {5462, 13367, 45735}, {5611, 5615, 41169}, {5640, 11464, 12106}, {5944, 10095, 3518}, {7503, 12161, 18436}, {7526, 7592, 34783}, {8254, 43575, 5}, {9730, 11430, 3}, {9818, 11402, 18445}, {9818, 18445, 18435}, {10564, 16836, 3}, {11425, 36752, 3}, {11426, 37506, 18438}, {11438, 39242, 3}, {12006, 43394, 22467}, {12022, 12228, 49}, {12022, 14389, 5}, {13336, 13346, 3}, {13339, 13353, 182}, {13339, 37472, 37477}, {13339, 37477, 3}, {13352, 37477, 37495}, {13353, 37472, 3}, {13353, 37477, 13339}, {14130, 43845, 185}, {15026, 32171, 44802}, {15062, 43602, 45957}, {34148, 43651, 140}, {36747, 37476, 3}, {36749, 37494, 1351}, {37471, 37495, 3}, {37472, 37477, 13352}, {37480, 44490, 37489}