HYACINTHOS 28074

 

 [César Lozada]:

 

Let ABC be a triangle, P a point and A’B’C’ the symmetric triangle of P, ie., A’, B’, C’ are the reflections of P in A, B, C, resp. Circles (ABB’) and (ACC’) meet again in A” and  B”, C” are defined cyclically. Then ABC and A”B”C” are perspective.

If P=x:y:z (barycentrics) then the perspector (ABC, A”B”C”)=Q(P) is:

Q(P) = (y*z*a^2-(b^2*z+c^2*y)*(x+2*y+2*z))*x : :

 

If P lies on the circumcircle the Q(P) = isogonal conjugate of P

If P lies on the line at infinity the Q(P) = P

 

ETC pairs (P,Q(P)): (1,3679), (3,3), (4,2)

 

Some others:

 

Q( X(2) ) = ISOTOMIC CONJUGATE OF X(18842)

= a^2-5*b^2-5*c^2 : :  (barys)

= 5*X(2)-2*X(6), 2*X(2)+X(69), X(2)-4*X(141), 7*X(2)-X(193), 7*X(2)-4*X(597), X(2)+2*X(599), 4*X(2)-X(1992), 11*X(2)-8*X(3589), 8*X(2)-5*X(3618), 4*X(2)-7*X(3619), X(2)+5*X(3620), 19*X(2)-4*X(3629), 7*X(2)+8*X(3631), 7*X(2)-10*X(3763), 31*X(2)-16*X(6329), 13*X(2)-4*X(8584), 16*X(2)-X(11008), 5*X(2)+X(11160), 7*X(2)+2*X(15533)

= on lines: {2, 6}, {3, 11147}, {4, 7883}, {7, 17228}, {8, 17227}, {20, 11164}, {30, 10519}, {67, 9143}, {76, 5485}, {144, 17285}, {145, 17305}, {182, 15702}, {315, 598}, {344, 17272}, {346, 17273}, {376, 1352}, {487, 6449}, {488, 6450}, {511, 3545}, {518, 4731}, {542, 3524}, {543, 7865}, {547, 1351}, {549, 6776}, {575, 3533}, {576, 5067}, {620, 8593}, {626, 7617}, {631, 7870}, {671, 19662}, {903, 4437}, {1153, 3788}, {1350, 3543}, {1353, 10124}, {1444, 16431}, {1494, 15595}, {1503, 10304}, {2345, 17288}, {2393, 7998}, {2482, 11161}, {2548, 14762}, {3090, 5476}, {3098, 11001}, {3161, 17329}, {3241, 3416}, {3448, 5648}, {3523, 15069}, {3525, 7909}, {3529, 7936}, {3564, 5054}, {3616, 17387}, {3672, 17295}, {3751, 3828}, {3785, 8369}, {3818, 15682}, {3830, 18358}, {3839, 10516}, {3849, 5207}, {3912, 16676}, {3917, 11188}, {3926, 8359}, {3934, 8176}, {4000, 17287}, {4357, 16673}, {4419, 17230}, {4644, 17292}, {4663, 19877}, {4748, 17244}, {4916, 17396}, {5050, 11539}, {5055, 14853}, {5056, 11477}, {5071, 20423}, {5085, 15708}, {5092, 15719}, {5181, 9140}, {5182, 9167}, {5206, 7795}, {5222, 17360}, {5296, 17241}, {5308, 17250}, {5343, 11304}, {5344, 11303}, {5461, 10754}, {5569, 7880}, {5642, 11061}, {5650, 8681}, {5749, 17361}, {5839, 17291}, {5921, 15692}, {6030, 11206}, {6172, 17342}, {6337, 7618}, {6393, 11287}, {6453, 11292}, {6454, 11291}, {7492, 19596}, {7615, 9466}, {7619, 7815}, {7620, 7841}, {7757, 14994}, {7767, 19661}, {7768, 16045}, {7776, 8367}, {7794, 16043}, {7811, 14039}, {7849, 14064}, {7854, 14001}, {7879, 8370}, {7937, 11054}, {8542, 11416}, {8550, 10303}, {8591, 11646}, {8703, 18440}, {10385, 12589}, {11050, 12583}, {11318, 16509}, {11812, 12017}, {11898, 15694}, {14645, 14971}, {14848, 15699}, {14912, 15709}, {15585, 20079}, {15703, 18583}, {15705, 21167}, {16475, 19883}, {17132, 17274}, {17133, 17294}, {17231, 17257}, {17236, 17314}, {17237, 17316}, {17269, 20073}, {17270, 21255}, {17289, 21296}, {17296, 17321}, {17328, 18230}

= reflection of X(i) in X(j) for these (i,j): (3839, 10516), (5050, 11539), (5182, 9167), (14848, 15699), (16475, 19883)

= complement of X(5032)

= isotomic conjugate of X(18842)

= X(3524)-of-anti-Artzt triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 69, 1992), (2, 1271, 13637), (2, 7840, 7736), (2, 11160, 6), (69, 141, 3619), (69, 3618, 11008), (69, 3619, 3618), (141, 3631, 3763), (298, 299, 15589), (597, 3631, 15533), (597, 15533, 193), (599, 15533, 3631), (3631, 3763, 193), (3763, 15533, 597), (7840, 16986, 2), (9761, 9763, 13468)

= [ 3.7203423241040830, 1.9286123854773510, 0.5883902192981747 ]

 

Q( X(5) ) = X(5)X(51) ∩ X(54)X(632)

= ((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-5*(b^2+c^2)*a^4+7*(b^4+b^2*c^2+c^4)*a^2-3*(b^4-c^4)*(b^2-c^2)) : :  (barys)

= (S^2+SB*SC)*(4*SA+2*SW-3*R^2) : : (barys)

= X(5)-4*X(1209), 7*X(5)-4*X(3574), 5*X(5)-8*X(13565), 5*X(5)-2*X(20424), X(5)+2*X(21230), 7*X(1209)-X(3574), 5*X(1209)-2*X(13565), 10*X(1209)-X(20424), 2*X(1209)+X(21230), 5*X(3574)-14*X(13565), 10*X(3574)-7*X(20424), 2*X(3574)+7*X(21230), 4*X(13565)-X(20424), 4*X(13565)+5*X(21230), X(20424)+5*X(21230)

= on lines: {5, 51}, {54, 632}, {140, 2888}, {195, 3628}, {539, 11539}, {546, 12307}, {547, 7605}, {549, 9140}, {550, 6288}, {597, 5965}, {1216, 13368}, {1483, 12785}, {1493, 15605}, {1656, 12325}, {3090, 12316}, {3519, 8254}, {3530, 12254}, {3627, 7691}, {3858, 20584}, {6101, 6153}, {6286, 10593}, {6343, 13372}, {7356, 10592}, {8703, 18400}, {10203, 18350}, {10386, 12956}, {10610, 14869}, {12013, 15699}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1209, 21230, 5), (13565, 20424, 5)

= [ 9.1996197060782810, -3.5577199859287820, 1.8577230693604840 ]

 

Q( X(6) ) = MIDPOINT OF X(3545) AND X(10519)

= a^2+4*b^2+4*c^2 : :  (barys)

= 4*X(2)-X(6), 5*X(2)+X(69), X(2)+2*X(141), 13*X(2)-X(193), 5*X(2)-2*X(597), 2*X(2)+X(599), 7*X(2)-X(1992), 7*X(2)-4*X(3589), 11*X(2)-5*X(3618), X(2)-7*X(3619), 7*X(2)+5*X(3620), 17*X(2)-2*X(3629), 11*X(2)+4*X(3631), 2*X(2)-5*X(3763), 5*X(2)-X(5032), 23*X(2)-8*X(6329), 11*X(2)-2*X(8584), 11*X(2)+X(11160), 8*X(2)+X(15533), 10*X(2)-X(15534)

= on lines: {2, 6}, {3, 11178}, {30, 10516}, {45, 17237}, {67, 5642}, {76, 10302}, {125, 5646}, {182, 9703}, {381, 1350}, {487, 6429}, {488, 6430}, {511, 5055}, {518, 3921}, {542, 5054}, {543, 11287}, {547, 20423}, {549, 1352}, {551, 3416}, {576, 5070}, {598, 7770}, {626, 8176}, {631, 11180}, {635, 11306}, {636, 11305}, {671, 7937}, {1078, 8366}, {1153, 7815}, {1351, 15703}, {1503, 3524}, {1656, 5476}, {1691, 5215}, {2023, 5503}, {2076, 3849}, {2393, 5650}, {2482, 11646}, {2930, 6698}, {2979, 16776}, {3053, 7810}, {3066, 15360}, {3094, 9466}, {3096, 7841}, {3098, 3830}, {3242, 3679}, {3525, 8550}, {3526, 10168}, {3534, 3818}, {3545, 10519}, {3564, 11539}, {3642, 11295}, {3643, 11296}, {3661, 17119}, {3662, 17118}, {3734, 5077}, {3751, 19876}, {3788, 7619}, {3834, 17308}, {3912, 16672}, {3917, 9971}, {3934, 7617}, {4265, 16418}, {4357, 16675}, {4361, 17228}, {4363, 17227}, {4389, 17269}, {4445, 16706}, {4657, 17311}, {4995, 12589}, {5013, 6292}, {5017, 7818}, {5023, 7800}, {5026, 10488}, {5064, 7716}, {5071, 5480}, {5092, 15701}, {5096, 16417}, {5102, 14848}, {5116, 7622}, {5152, 9830}, {5207, 14036}, {5210, 7820}, {5254, 5485}, {5298, 12588}, {5463, 11481}, {5464, 11480}, {5569, 11288}, {5651, 18374}, {5847, 19883}, {5921, 15721}, {5969, 9166}, {6034, 14971}, {6379, 9462}, {6593, 13169}, {6697, 9924}, {6776, 15702}, {7232, 17289}, {7603, 11173}, {7618, 7795}, {7738, 11148}, {7746, 10542}, {7757, 10007}, {7761, 11159}, {7775, 7849}, {7784, 8370}, {7812, 7879}, {7814, 9731}, {7817, 7914}, {7819, 19661}, {7833, 11164}, {7845, 14535}, {7870, 11285}, {7998, 9019}, {8288, 10717}, {8360, 13881}, {8681, 15082}, {8703, 18358}, {9813, 10510}, {10150, 15514}, {10304, 21167}, {10387, 11238}, {10719, 15162}, {10720, 15163}, {11049, 12583}, {11284, 19510}, {11301, 13083}, {11302, 13084}, {11898, 15723}, {14046, 18906}, {14561, 15699}, {14810, 15681}, {15303, 16176}, {15693, 18440}, {15707, 17508}, {16041, 20112}, {16777, 17231}, {16884, 17296}, {16885, 17272}, {17132, 17281}, {17133, 17301}, {17229, 17304}, {17230, 17305}, {17233, 17323}, {17235, 17286}, {17236, 17262}, {17239, 17282}, {17240, 17324}, {17241, 17326}, {17249, 17268}, {17250, 17266}, {17252, 17341}, {17253, 17279}, {17254, 17342}, {17255, 17280}, {17270, 17356}, {17273, 17358}, {17287, 17370}, {17288, 17371}, {17294, 17382}, {17295, 17383}, {17298, 17385}, {17302, 17309}, {17303, 21255}, {17310, 17399}, {17312, 17400}

= midpoint of X(3545) and X(10519)

= reflection of X(i) in X(j) for these (i,j): (1691, 5215), (5102, 14848), (6034, 14971), (10304, 21167), (14561, 15699)

= X(5054)-of-anti-Artzt triangle

= X(9166)-of-1st Brocard triangle

= X(11287)-of-Artzt triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 69, 597), (2, 491, 13663), (2, 492, 13783), (2, 7840, 11174), (2, 11160, 3618), (69, 597, 15534), (599, 3763, 2), (3618, 3631, 6144), (3618, 11160, 8584), (3631, 8584, 11160), (7840, 16988, 2), (8252, 8253, 3054), (8584, 11160, 6144), (9761, 9763, 8667), (16644, 16645, 230), (17232, 17307, 15668)

= [ 3.1748555582964020, 1.8407624569015810, 0.9009725233003298 ]

 

Q( X(13) ) = X(2)X(18) ∩ X(13)X(298)

= 5*S^2+sqrt(3)*(3*SA+SW)*S+3*SB*SC : :  (barys)

= X(13)+2*X(298), X(13)-4*X(623), X(298)+2*X(623), 2*X(316)+X(6779), 2*X(616)+X(19106), 2*X(618)+X(621), 4*X(620)-X(6780), X(3180)-4*X(6669), 2*X(3180)-5*X(16960), X(5473)+2*X(20428), 2*X(5978)+X(6777), 8*X(6669)-5*X(16960)

= on lines: {2, 18}, {13, 298}, {14, 11297}, {17, 5859}, {30, 5463}, {299, 16966}, {302, 3642}, {316, 6779}, {381, 5864}, {524, 16267}, {599, 1351}, {616, 19106}, {618, 621}, {620, 6780}, {754, 9761}, {3180, 6669}, {3411, 11289}, {3545, 7685}, {3643, 16809}, {5054, 5464}, {5858, 7759}, {5978, 6777}, {7860, 11303}, {11299, 16964}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (298, 623, 13), (302, 3642, 16242), (3180, 6669, 16960), (9761, 11298, 16963)

= [ 3.1651265206060310, 1.4173454059835760, 1.1985977297929660 ]

 

Q( X(14) ) = X(2)X(17) ∩ X(14)X(299)

= 5*S^2-sqrt(3)*(3*SA+SW)*S+3*SB*SC : :  (barys)

= X(14)+2*X(299), X(14)-4*X(624), X(299)+2*X(624), 2*X(316)+X(6780), 2*X(617)+X(19107), 2*X(619)+X(622), 4*X(620)-X(6779), X(3181)-4*X(6670), 2*X(3181)-5*X(16961), X(5474)+2*X(20429), 2*X(5979)+X(6778), 8*X(6670)-5*X(16961)

= on lines: {2, 17}, {13, 11298}, {14, 299}, {18, 5858}, {30, 5464}, {298, 16967}, {303, 3643}, {316, 6780}, {381, 5865}, {524, 16268}, {599, 1351}, {617, 19107}, {619, 622}, {620, 6779}, {754, 9763}, {3181, 6670}, {3412, 11290}, {3545, 7684}, {3642, 16808}, {5054, 5463}, {5859, 7759}, {5979, 6778}, {7860, 11304}, {11300, 16965}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (299, 624, 14), (303, 3643, 16241), (3181, 6670, 16961), (9763, 11297, 16962)

= [ 8.3237346581989260, 4.2251969033108120, -3.1261955241687650 ]

 

César Lozada