Let ABC be a triangle. A'B'C' the pedal triangle of H, P a point, A"B"C" the pedal triangle of P and L = the Euler line.
Denote:
A2, B2, C2 = the orthogonal projections of A", B", C" on A'A1, B'B1, C'C1, resp.
(or
Aa, Bb, Cc = the orthogonal projections of A', B', C' on L, resp.
A1, B1, C1 = the reflections of Aa, Bb, Cc in BC, CA, AB, resp.
A2, B2, C2 = the orthogonal projections of A", B", C" on A'A1, B'B1, C'C1, resp.)
O* = the circumcenter of A2B2C2.
Which is the locus of P such that O* lies on the Euler line?
The Euler line?
[César Lozada]:
Locus = {Euler line} ∪ {a ugly conic q2}
q2 = ∑ [(SB+SC)*((-5*R^2+SA+SW)*S^4+( 864*R^6+(324*SA-648*SW)*R^4+(- 21*SA^2-132*SA*SW+160*SW^2)*R^ 2+(4*SA^2+13*SA*SW-13*SW^2)* SW)*S^2-2*(4*R^2-SW)*(36*R^2- 7*SW)*SA^2*SW)*x^2+2*S^2*(S^4+ (294*R^4+(2*SA-114*SW)*R^2+10* SW^2)*S^2+(-864*SA+576*SW)*R^ 6+8*(27*SA^2+45*SA*SW-53*SW^2) *R^4-2*(42*SA^2+19*SA*SW-49* SW^2)*SW*R^2+(8*SA^2-7*SW^2)* SW^2)*z*y ] = 0 (barys)
No ETC centers on q2. Center and perspector of q2 not interesting.
For P on the Euler line, if OP/OH=t then OO*=((t+1)/4)*OH, ie., O* = homothecy of G with center P and radius 3/4.
O* = 3*S^2-SB*SC -(S^2-3*SB*SC)*t : : (barys) = S^2*(3-t)+(3*t-1)*SB*SC : :
= (t-3)*X(3)-(t+1)*X(4)
ETC-pairs (P,O*):
(2,2), (3,140), (4,5), (5,3628), (20,3), (21,6675), (22,6676), (23,468), (25,6677), (26,10020), (27,6678), (376,549), (377,8728), (381,547), (382,546), (384,7819), (401,441), (428,10128), (452,11108), (548,12108), (549,10124), (550,3530), (631,632), (858,5159), (1003,8368), (1370,1368), (1513,10011), (1650,15184), (1657,548), (1658,10125), (2071,10257), (2475,442), (3091,1656), (3146,4), (3151,440), (3153,2072), (3522,631), (3523,3526), (3524,11539), (3528,14869), (3529,550), (3534,12100), (3543,381), (3545,15699), (3552,7807), (3575,9825), (3627,3850), (3830,5066), (3832,3090), (3839,5055), (3843,12812), (3845,10109), (3853,12811), (3854,7486), (4188,13747), (4189,7483), (4190,474), (4198,7535), (4240,402), (5025,8361), (5046,4187), (5056,5070), (5059,20), (5068,5067), (5073,3853), (5133,11548), (5189,858), (5498,12043), (5501,12056), (5899,10096), (6636,7499), (6655,6656), (6656,8364), (6658,384), (6836,6922), (6837,6861), (6838,6863), (6839,6881), (6840,6882), (6847,6862), (6848,6959), (6872,405), (6890,6958), (6895,6831), (6925,6907), (6934,6924), (6938,6914), (6987,6883), (6994,7522), (6995,5020), (7387,13383), (7391,427), (7408,7392), (7464,15122), (7471,12068), (7487,6642), (7488,7542), (7492,7495), (7500,25), (7512,7568), (7518,7532), (7519,1995), (7520,7561), (7538,7515), (7544,7405), (7560,7536), (7576,10127), (7667,7734), (7791,8362), (7833,8359), (7841,8360), (7933,8363), (8352,8355), (8353,8358), (8369,8365), (8370,8367), (8703,11812), (10126,12057), (10296,10297), (10304,5054), (10431,8727), (11001,8703), (11050,11049), (11250,5498), (11413,16196), (11414,16197), (11563,15350), (12100,11540), (12225,12362), (13619,15646), (13743,10021), (14035,7770), (14063,7887), (14142,15327), (14790,13371), (14807,1313), (14808,1312), (14953,1375), (15640,3830), (15677,15670), (15678,15673), (15680,21), (15682,3845), (15683,376), (15684,14893), (15685,15690), (15686,14891), (15687,11737), (15692,15694), (15697,15693), (15705,15709), (15717,3525), (15721,15723), (15971,15973), (16117,11277)
Examples:
O*(X(24)) = complement of X(11585)
= (14*R^2-3*SW)*S^2-(2*R^2-SW)* SB*SC : : (barys)
= 3*X(2)+X(24), X(4)+3*X(15078)
= As a point on the Euler line, this center has Shinagawa coefficients (E-6*F, E+2*F)
= on lines: {2, 3}, {49, 11245}, {52, 11064}, {125, 12134}, {389, 5972}, {1147, 13292}, {1493, 5181}, {1511, 12370}, {3589, 6153}, {5446, 14156}, {5448, 13568}, {5654, 9786}, {6696, 6699}, {8263, 8548}, {9306, 12359}, {10272, 14708}, {10280, 14341}, {11449, 12022}, {12038, 12241}, {13336, 13394}, {14389, 15024}, {14984, 15120}
= complement of X(11585)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2, 24, 11585), (3, 468, 13383), (3, 1656, 6816), (5, 549, 7526), (5, 6642, 10127), (140, 10020, 6676), (140, 10096, 548), (140, 13383, 3), (468, 7499, 7493), (468, 11799, 10096), (858, 3518, 7553), (3546, 6353, 7387), (5159, 6756, 13371), (6640, 7506, 427), (6643, 14070, 550), (12106, 13371, 6756)
= [ 2.3698180729645410, 1.4940465101533110, 1.5125624027407610 ]
O*( X(140) ) = complement of X(3628)
= b*c*(6*a^4-11*(b^2+c^2)*a^2+5* (b^2-c^2)^2) : : (trilinears)
= 11*S^2-SB*SC : : (barys)
= As a point on the Euler line, this center has Shinagawa coefficients (11, -1)
= 15*X(2)+X(3), 9*X(2)-X(5), 3*X(2)+X(140), 17*X(2)-X(381), 21*X(2)-X(546), 5*X(2)-X(547), 7*X(2)+X(549), 3*X(2)+5*X(632), 21*X(2)-5*X(1656), 9*X(2)+7*X(3526), 9*X(2)+X(3530), 15*X(2)-X(3850), 18*X(2)-X(3856), 19*X(2)-X(3860), 13*X(2)+3*X(5054), 19*X(2)-3*X(5055), 13*X(2)-X(5066), 27*X(2)-11*X(5070), 7*X(2)-X(10109), 5*X(2)+3*X(11539), 2*X(2)+X(11540), 11*X(2)-X(11737), 5*X(2)+X(11812), 11*X(2)+X(12100), 6*X(2)+X(12108), 12*X(2)-X(12811), 11*X(2)+3*X(14890), 13*X(2)+X(14891), 11*X(2)+5*X(15694), 11*X(2)-3*X(15699), 23*X(2)-7*X(15703), 5*X(2)+11*X(15723), 17*X(2)+X(15759), 11*X(3)+5*X(4), 3*X(3)+5*X(5), 21*X(3)-5*X(20), X(3)-5*X(140), 31*X(3)-15*X(376), 7*X(3)+5*X(546), X(3)+3*X(547), 9*X(3)-5*X(548), 7*X(3)-15*X(549), 13*X(3)-5*X(550), 3*X(3)-5*X(3530), X(3)-17*X(3533), 13*X(3)+3*X(3543), 7*X(3)+9*X(3545), X(3)+5*X(3628)
= on lines: {2, 3}, {125, 13392}, {143, 3819}, {230, 5041}, {323, 15047}, {373, 10263}, {395, 3412}, {396, 3411}, {485, 6438}, {486, 6437}, {952, 3634}, {1125, 5844}, {1131, 6446}, {1132, 6445}, {1154, 11695}, {1216, 13363}, {1353, 3619}, {1483, 9780}, {3054, 5305}, {3055, 5008}, {3070, 6485}, {3071, 6484}, {3316, 6395}, {3317, 6199}, {3589, 5097}, {3592, 10194}, {3594, 10195}, {3624, 5690}, {3828, 15178}, {3917, 14449}, {4301, 11230}, {5432, 15172}, {5447, 6688}, {5462, 15606}, {5550, 10283}, {5609, 13393}, {5650, 6101}, {5704, 15935}, {5843, 6666}, {5886, 9588}, {5892, 11591}, {5893, 10193}, {5901, 11231}, {5943, 10627}, {5946, 14531}, {6243, 11465}, {6429, 9680}, {6431, 8252}, {6432, 8253}, {6433, 9681}, {6668, 6681}, {6671, 6674}, {6672, 6673}, {6689, 15605}, {6704, 14693}, {7294, 15325}, {7746, 9607}, {7751, 15597}, {7759, 9771}, {7767, 7814}, {7888, 11168}, {8254, 11064}, {8972, 13961}, {9624, 11531}, {9657, 10592}, {9670, 10593}, {9693, 10137}, {9706, 13353}, {9729, 14128}, {10110, 12045}, {10170, 13630}, {10172, 13624}, {10219, 13391}, {10272, 16003}, {10386, 10589}, {10625, 13451}, {11017, 14915}, {11793, 12006}, {13364, 15644}, {13903, 13941}, {14643, 15057}
= midpoint of X(i) and X(j) for these {i,j}: {2, 10124}, {125, 13392}, {5447, 10095}, {5609, 13393}, {9729, 14128}, {11793, 12006}
= complement of X(3628)
= X(3856) of Johnson triangle
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3, 3543, 550), (4, 12812, 11737), (4, 15699, 12812), (5, 382, 3859), (5, 3627, 3855), (5, 3853, 3850), (382, 3859, 3861), (382, 5070, 7486), (382, 7486, 5), (547, 3853, 5), (1656, 3830, 15022), (2041, 2042, 5054), (3627, 15713, 3523), (3628, 12102, 3090), (3832, 3853, 3861), (12102, 14891, 550)
= [ 2.8889309238454450, 2.0117899282261620, 0.9145341052068311 ]
César Lozada
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