VARIATIONS (of Hyacinthos 28502)
Let ABC be a triangle. P a point and A'B'C' :1. the pedal triangle of O,
or
2. the pedal triangle of P.
Denote:
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
D = the Poncelet point of ABCP
DA' intersects again (Nb), (Nc) at Ab, Ac, resp.
DB' intersects again (Nc), (Na) at Bc, Ba, resp.
DC' intersects again (Na), (Nb) at Ca, Cb resp.
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
1. Ma, Mb, Mc and D are concyclic.
Centers (for the two cases of A'B'C')?
2.The perpendicular bisectors of AbAc, BcBa, CaCb, are concurrent at the antipode of D on the circle (MaMbMcD)
Points (for the two cases of A'B'C')?
[César Lozada]:
2)
For A’B’C’ = pedal triangle of P.
For P=x:y:z (barys), the center of the circle is
O*(P) = x^3*(SB*c^4*y^3+SC*b^4*z^3)-(5*S^2-(2*SB+SC)*SA)*c^2*x^3*y^2*z-(5*S^2-(SB+2*SC)*SA)*b^2*x^3*y*z^2-(2*S^2-(SB-SC)*SB)*c^2*x^2*y^3*z-4*SW*(S^2+SB*SC)*x^2*y^2*z^2-(2*S^2+(SB-SC)*SC)*b^2*x^2*y*z^3-(S^2+(SB+2*SC)*SB)*a^2*x*y^3*z^2-(S^2+(2*SB+SC)*SC)*a^2*x*y^2*z^3-a^6*y^3*z^3 : :
and the given perpendicular bisectors concur at:
D*(P) = SB*c^8*x^5*y^5+(-3*S^2+(2*SB-SC)*SA)*c^6*x^5*y^4*z+(2*S^4-(14*SA-SB+SC)*SA*S^2+(5*SB+3*SC)*SA^3)*c^2*x^5*y^3*z^2+(2*S^4-(14*SA+SB-SC)*SA*S^2+(3*SB+5*SC)*SA^3)*b^2*x^5*y^2*z^3+(-3*S^2-(SB-2*SC)*SA)*b^6*x^5*y*z^4+SC*b^8*x^5*z^5+(-2*S^2+(3*SB+SC)*SB)*c^6*x^4*y^5*z+(4*S^4+(-14*SA^2-10*SB^2+2*SB*SC)*S^2+2*(3*SA^2*SB+3*SA^2*SC+SA*SB^2+4*SB^3)*SA)*c^2*x^4*y^4*z^2+((-16*SA+2*SB+2*SC)*S^4-(45*SA^2-40*SB*SC-3*SW^2)*SA*S^2-3*(5*SA^2-6*SB*SC-5*SW^2)*SA^3)*x^4*y^3*z^3+(4*S^4+(-14*SA^2+2*SB*SC-10*SC^2)*S^2+2*(3*SA^2*SB+3*SA^2*SC+SA*SC^2+4*SC^3)*SA)*b^2*x^4*y^2*z^4+(-2*S^2+(SB+3*SC)*SC)*b^6*x^4*y*z^5-(-4*S^4+(3*SA^2-2*SA*SB+SA*SC+2*SB^2)*S^2-3*SA^3*SB+3*SA^2*SB^2-3*SA*SB^3-2*SB^4-3*SA^3*SC)*c^2*x^3*y^5*z^2+((38*SA-6*SB-2*SC)*S^4+(42*SA^3-56*SA^2*SB-44*SA^2*SC+23*SA*SB^2+3*SA*SC^2-12*SB^3)*S^2-3*(14*SA^3*SB+14*SA^3*SC-9*SA^2*SB^2-5*SA^2*SC^2+4*SA*SB^3-4*SB^4)*SA)*x^3*y^4*z^3+((38*SA-2*SB-6*SC)*S^4+(42*SA^3-44*SA^2*SB-56*SA^2*SC+3*SA*SB^2+23*SA*SC^2-12*SC^3)*S^2-3*(14*SA^3*SB+14*SA^3*SC-5*SA^2*SB^2-9*SA^2*SC^2+4*SA*SC^3-4*SC^4)*SA)*x^3*y^3*z^4-(-4*S^4+(3*SA^2+SA*SB-2*SA*SC+2*SC^2)*S^2-3*SA^3*SB+3*SA^2*SC^2-3*SA*SC^3-2*SC^4-3*SA^3*SC)*b^2*x^3*y^2*z^5-((SA+2*SB)*S^2-(3*SA-2*SB)*SB^2)*a^4*x^2*y^5*z^3-2*((9*SA+3*SB+3*SC)*S^2+(4*SA^2-5*SB*SC-4*SW^2)*SA)*a^4*x^2*y^4*z^4-((SA+2*SC)*S^2-(3*SA-2*SC)*SC^2)*a^4*x^2*y^3*z^5-(-S^2+(3*SB+2*SC)*SB)*a^6*x*y^5*z^4-(-S^2+(2*SB+3*SC)*SC)*a^6*x*y^4*z^5-a^10*y^5*z^5 : :
D*(P) is indeed the antipode of D in the above circle.
ETC pairs (P, O*(P)): (1,5), (4,5), (84,10943)
ETC pairs (P, D*(P)): (1,119)
Some others:
O*(X(2)) = MIDPOINT OF X(2) AND X(3818)
= 2*a^6+(b^2+c^2)*a^4+(b^4+12*b^2*c^2+c^4)*a^2-4*(b^4-c^4)*(b^2-c^2) : : (barys)
= 4*X(5)-X(575), 3*X(5)-X(597), 7*X(5)-X(8550), 2*X(5)+X(18553), X(6)-5*X(19709), 3*X(575)-4*X(597), 7*X(575)-4*X(8550), X(575)+2*X(18553), 7*X(597)-3*X(8550), 2*X(597)+3*X(18553), 2*X(3818)+X(5092), 2*X(8550)+7*X(18553), 3*X(14810)-8*X(20582), X(14810)-4*X(24206), 2*X(20582)-3*X(24206)
= on lines: {2, 1495}, {5, 542}, {6, 18362}, {30, 14810}, {141, 3845}, {182, 5055}, {373, 9140}, {381, 511}, {524, 5066}, {547, 1503}, {576, 3851}, {671, 3399}, {1350, 14269}, {1352, 1992}, {1656, 20190}, {1853, 10219}, {3091, 7946}, {3098, 3830}, {3363, 19662}, {3534, 3763}, {3564, 11737}, {3589, 10109}, {3619, 15682}, {3849, 9996}, {5031, 7880}, {5071, 11179}, {5072, 14848}, {5085, 15703}, {5169, 13857}, {5650, 10989}, {6287, 13335}, {7533, 15360}, {7603, 11646}, {7809, 14994}, {7841, 10356}, {9041, 18357}, {10706, 15030}, {11180, 14561}, {11182, 18309}, {11801, 20113}, {14892, 18583}, {14927, 15709}, {15060, 16776}, {15686, 21167}, {15694, 17508}, {16626, 22579}, {16627, 22580}, {21850, 22165}
= midpoint of X(i) and X(j) for these {i,j}: {2, 3818}, {141, 3845}, {381, 11178}, {1352, 5476}, {3098, 3830}, {5066, 18358}, {11182, 18309}, {15060, 16776}, {21850, 22165}
= reflection of X(i) in X(j) for these (i,j): (3589, 10109), (5092, 2), (5097, 5476), (10168, 547), (19130, 5066)
= X(22566)-of-1st Brocard triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 18553, 575), (381, 10516, 11178), (1352, 3545, 5476), (3830, 21358, 3098), (5459, 5460, 7817)
= [ -0.0762617094362402, -1.2877288240465550, 4.5673667644487200 ]
D*(X(2)) = REFLECTION OF X(115) IN O*(X(2))
= 3*(5*SW+3*SA)*S^4+(9*SA^2-18*SA*SW+2*SW^2)*SW*S^2-9*SB*SC*SW^3 : : (barys)
= 3*X(114)-X(18800), 3*X(1352)-X(11161), 2*X(5092)-3*X(9167), 3*X(6034)-5*X(19709), 3*X(6054)+X(11161), 3*X(10516)-X(11632), 3*X(11178)-2*X(19662), X(11179)-3*X(23234), X(14830)-3*X(21358), X(14981)+2*X(18553)
= on lines: {2, 98}, {316, 19924}, {524, 22566}, {543, 3818}, {599, 6033}, {690, 18309}, {2482, 11645}, {3845, 5969}, {5024, 11646}, {5092, 9167}, {6034, 19709}, {9830, 12040}, {10516, 11632}, {12042, 20582}, {14830, 21358}, {14981, 18553}
= midpoint of X(i) and X(j) for these {i,j}: {147, 19905}, {599, 6033}, {1352, 6054}, {11180, 12177}
= reflection of X(i) in X(j) for these (i,j): (6055, 24206), (12042, 20582), (115, O*(X(2)) )
= [ -2.3772118807815010, -5..9632429616871750, 8.8662381695900390 ]
---------------------------------
O*(X(3)) = MIDPOINT OF X(3) AND X(20299)
= 2*a^10-4*(b^2+c^2)*a^8-(b^4-12*b^2*c^2+c^4)*a^6+7*(b^4-c^4)*(b^2-c^2)*a^4-(b^2-c^2)^2*(5*b^4+8*b^2*c^2+5*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : : (barys)
= (28*R^2-SA-6*SW)*S^2-(20*R^2-3*SW)*SB*SC : : (barys)
= 3*X(2)+X(3357), 9*X(2)-X(5878), 15*X(2)+X(12250), 5*X(3)+3*X(1853), X(3)-3*X(10193), 9*X(3)-X(17845), 3*X(3)+X(18381), X(3)+3*X(23329), X(1853)+5*X(10193), 9*X(1853)-5*X(18381), 3*X(1853)-5*X(20299), X(1853)-5*X(23329), 3*X(3357)+X(5878), 5*X(3357)-X(12250), 5*X(5878)+3*X(12250), X(6288)-3*X(14076), 9*X(10193)+X(18381), 3*X(10193)+X(20299), X(17845)+3*X(18381), X(17845)+9*X(20299), X(18381)-3*X(20299), X(18381)-9*X(23329), X(20299)-3*X(23329)
= on lines: {2, 3357}, {3, 161}, {4, 11204}, {5, 1539}, {20, 23325}, {30, 20191}, {49, 16003}, {54, 5900}, {64, 3526}, {66, 17508}, {74, 6143}, {125, 3520}, {140, 6000}, {154, 15720}, {185, 10294}, {186, 13419}, {381, 8567}, {468, 13474}, {542, 12038}, {547, 5893}, {549, 6247}, {550, 18383}, {578, 18916}, {631, 5651}, {632, 2883}, {1204, 18388}, {1216, 15122}, {1498, 5054}, {1503, 3530}, {1568, 11440}, {1594, 21663}, {1614, 13399}, {1620, 18494}, {1656, 10606}, {1657, 18376}, {1658, 6697}, {2781, 12006}, {3516, 18390}, {3521, 15041}, {3523, 11202}, {3524, 9833}, {3533, 6225}, {3541, 11438}, {3628, 15311}, {3851, 5925}, {4550, 18431}, {5055, 5895}, {5449, 11250}, {5498, 5663}, {5876, 14156}, {5890, 12242}, {5907, 10257}, {5972, 12162}, {7577, 11468}, {7687, 18560}, {7689, 18281}, {8254, 10628}, {8981, 13980}, {8991, 13966}, {9705, 12317}, {9919, 22462}, {10018, 11381}, {10116, 10264}, {10192, 14869}, {10226, 13561}, {10295, 11572}, {11231, 12262}, {11487, 22581}, {11793, 16196}, {12289, 23040}, {13093, 15694}, {13293, 14130}, {13382, 23292}, {13434, 15057}, {13754, 23336}, {14530, 15701}, {14810, 23300}, {14864, 15712}, {15055, 19506}, {15696, 18405}, {15704, 23324}, {21659, 23294}
= midpoint of X(i) and X(j) for these {i,j}: {3, 20299}, {140, 6696}, {550, 18383}, {5449, 11250}, {6247, 10282}, {6697, 15578}, {10193, 23329}, {10226, 13561}, {14810, 23300}
= complement of the complement of X(3357)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 23329, 20299), (125, 3520, 13403), (549, 6247, 10282), (550, 23332, 18383), (631, 6759, 10182), (1656, 10606, 22802), (3523, 14216, 11202), (10193, 20299, 3)
= [ 6.7732374643960500, 6.2035514047892350, -3.7802114742063470 ]
D*(X(3)) = REFLECTION OF X(125) IN O*(X(3))
= (SB+SC)*((5*R^2-SA-SW)*S^2-(3*R^2*(60*R^2-2*SA-25*SW)+SA^2-SB*SC+8*SW^2)*SA) : : (barys)
= 3*X(3)+X(2935), 9*X(3)-X(9919), 5*X(3)-X(10117), 3*X(3)-X(13289), X(74)-3*X(11204), X(265)-3*X(23329), X(399)+3*X(10606), X(1177)-3*X(17508), X(1498)-5*X(15040), 3*X(2935)+X(9919), 5*X(2935)+3*X(10117), 7*X(3528)+X(13203), 2*X(6699)-3*X(10193), X(6759)-3*X(15035), X(9934)-3*X(11202), X(9934)-5*X(15051), 3*X(11202)-5*X(15051)
= on lines: {3, 113}, {20, 19506}, {30, 15090}, {74, 184}, {110, 3357}, {125, 3520}, {186, 13202}, {265, 23329}, {378, 7687}, {399, 10606}, {542, 12901}, {550, 23315}, {974, 11430}, {1177, 17508}, {1204, 15463}, {1498, 15040}, {1511, 6000}, {1539, 15646}, {1614, 17856}, {1986, 21663}, {2071, 12827}, {2778, 13624}, {2781, 5092}, {3528, 13203}, {5010, 19505}, {5663, 10226}, {6699, 10193}, {6723, 7526}, {6759, 15035}, {7280, 10118}, {7712, 9934}, {7722, 11468}, {8546, 15578}, {8567, 10620}, {9976, 10249}, {10114, 20417}, {10264, 23328}, {10272, 15311}, {10605, 12227}, {10610, 10628}, {10721, 21844}, {10733, 23325}, {11250, 17702}, {11410, 19457}, {11413, 22109}, {11438, 15472}, {11454, 12219}, {12084, 12893}, {12121, 18381}, {12244, 23040}, {12292, 17701}, {13367, 17854}, {14130, 23515}, {14915, 20773}, {15041, 17847}, {15042, 17812}, {16219, 17835}, {18281, 19479}
= midpoint of X(i) and X(j) for these {i,j}: {3, 13293}, {20, 19506}, {110, 3357}, {550, 23315}, {1511, 11598}, {2935, 13289}, {12084, 12893}, {12121, 18381}
= reflection of X(125) in O*(X(3))
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 2935, 13289), (9934, 15051, 11202), (13289, 13293, 2935)
= [ 10.0562032271098700, 9.3273644209189900, -7.4580662220102460 ]
---------------------------------
O*(X(6)) = MIDPOINT OF X(2) AND X(19130)
= 2*a^6-8*(b^2+c^2)*a^4+(b^4-24*b^2*c^2+c^4)*a^2+5*(b^4-c^4)*(b^2-c^2) : : (barys)
= 5*X(2)-X(3098), 5*X(5)+X(575), 3*X(5)+X(597), 11*X(5)+X(8550), 7*X(5)-X(18553), X(182)+3*X(3545), 5*X(381)+3*X(5085), 3*X(575)-5*X(597), 11*X(575)-5*X(8550), 7*X(575)+5*X(18553), 11*X(597)-3*X(8550), 7*X(597)+3*X(18553), X(3098)+5*X(19130), 3*X(5085)-5*X(10168), 7*X(8550)+11*X(18553)
= on lines: {2, 3098}, {5, 542}, {182, 3545}, {373, 12824}, {381, 5085}, {511, 547}, {524, 10109}, {576, 5056}, {599, 1351}, {1503, 11737}, {1506, 6034}, {1992, 5071}, {3090, 20423}, {3543, 17508}, {3589, 5066}, {3818, 19709}, {3845, 5092}, {3850, 20190}, {5079, 14848}, {5480, 15699}, {7570, 15360}, {7605, 9140}, {7875, 10033}, {9830, 15092}, {11539, 14810}, {18583, 20583}
= midpoint of X(i) and X(j) for these {i,j}: {2, 19130}, {381, 10168}, {3589, 5066}, {3845, 5092}, {5476, 24206}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5055, 5476, 24206), (5071, 14561, 11178)
= [ 0.8674384671704768, 0.1667044057078022, 3.1248975238772180 ]
D*(X(6)) = REFLECTION OF X(125) IN O*(X(6))
= 12*S^4-(27*R^2*(SA-2*SW)-12*SA^2+12*SB*SC+16*SW^2)*S^2+3*(27*R^2-4*SW)*SB*SC*SW : : (barys)
= 3*X(113)+X(15303), 3*X(9970)+X(13169), X(9976)-3*X(14848), 3*X(11178)-X(13169)
= on lines: {6, 13}, {23, 5642}, {182, 10706}, {541, 10168}, {5972, 21766}, {6593, 11645}, {9970, 11178}
= midpoint of X(i) and X(j) for these {i,j}: {182, 10706}, {381, 19140}, {5476, 5655}, {9970, 11178}
= reflection of X(125) in O*(X(6))
= [ -1.7553947673412790, -2.7463295772438770, 6.3521517741568860 ]
César Lozada
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