Denote:
La, Lb, Lc = the reflections of L in BC, CA, AB, resp.
A*, B*, C* = the orthogonal projections of O on La, Lb, Lc, resp.
The triangles A'B'C', A*B*C* are Eulerologic.
ie
The Euler lines of A*B'C', B*C'A', C*A'B' are concurrent.
Which are the loci of the Eulerologic centers as L moves around H?
[César Lozada]:
Very hard to be solved for general case.
Eulerologic centers for two particular cases:
· Particular case 1: L = Euler line of ABC
A*->A’ = X(5)X(1986) ∩ X(113)X(389)
= (SB+SC)*((14*R^2-SA-3*SW)*S^2+ (6*R^2*(3*R^2+SA)-2*SW*(7*R^2- SW)-SA^2+SB*SC)*SA) : : (barys)
= X(3)+2*X(1112), X(3)-4*X(9826), X(4)+2*X(14708), 2*X(5)+X(1986), 4*X(5)-X(7723), X(52)+2*X(5972), X(74)-7*X(15043), X(110)+5*X(3567), X(110)+2*X(12236), X(113)+2*X(389), X(125)-4*X(5462), X(125)+2*X(11557), X(1112)+2*X(9826), 2*X(1986)+X(7723), 5*X(3567)-2*X(12236), 2*X(5462)+X(11557)
= on lines: {3, 1112}, {4, 14708}, {5, 1986}, {24, 12228}, {52, 5972}, {74, 15043}, {110, 3567}, {113, 389}, {125, 5462}, {143, 1511}, {265, 11746}, {381, 5640}, {399, 7529}, {546, 12292}, {569, 13289}, {973, 11597}, {974, 7728}, {1199, 3047}, {1539, 13630}, {1656, 12358}, {2070, 11416}, {2777, 9730}, {2781, 14561}, {3060, 15035}, {3090, 12219}, {3091, 7722}, {3448, 7528}, {3526, 13416}, {3843, 12133}, {3851, 13148}, {5093, 13321}, {5446, 16163}, {5562, 12900}, {5609, 13358}, {5644, 15041}, {5943, 10628}, {6102, 12825}, {6153, 14049}, {6644, 15463}, {6699, 13417}, {7403, 10264}, {7553, 11566}, {7687, 11562}, {7731, 15024}, {9729, 11807}, {9781, 10733}, {10020, 12606}, {10095, 10113}, {10110, 12295}, {10539, 12227}, {10574, 10721}, {10982, 12302}, {11424, 12901}, {11806, 15063}, {12006, 12041}, {13201, 15028}, {15045, 15055}
= midpoint of X(i) and X(j) for these {i,j}: {568, 14643}, {3060, 15035}
= X(2072) of orthocentroidal triangle
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (5, 1986, 7723), (1112, 9826, 3), (5462, 11557, 125), (7731, 15024, 15059), (9729, 11807, 16111), (10095, 11561, 10113)
= [ 1.2179893135455340, 0.7822439902789720, 2.5369620362317680 ]
A’->A* = X(5)X(110) ∩ X(110)X(389)
= (SB+SC)*(2*S^2*(5*R^2-SW)+(3* R^2*(12*R^2+SA)-SW*(19*R^2-2* SW))*SA) : : (barys)
= X(3)+2*X(11557), 2*X(3)+X(13417), X(5)+2*X(11561), 2*X(5)+X(11562), X(20)+2*X(11807), X(52)+2*X(1511), X(74)-4*X(9729), 2*X(113)+X(185), X(113)+2*X(14708), X(125)-4*X(9826), X(185)-4*X(14708), 2*X(974)+X(15063), X(10264)-4*X(12006), 4*X(11557)-X(13417), 4*X(11561)-X(11562)
= on lines: {2, 10628}, {3, 11557}, {5, 113}, {20, 11807}, {52, 1511}, {74, 9729}, {110, 389}, {146, 10574}, {182, 1205}, {186, 249}, {195, 568}, {265, 5462}, {399, 11806}, {631, 7731}, {1112, 16163}, {1539, 10575}, {1986, 5562}, {2777, 12824}, {3090, 12281}, {3091, 12270}, {3448, 15043}, {3523, 13201}, {3567, 11800}, {5446, 12121}, {5642, 14831}, {5892, 15061}, {5907, 7722}, {5943, 14644}, {6102, 10272}, {6240, 15473}, {6243, 15040}, {7723, 12900}, {9786, 12168}, {10110, 10733}, {10114, 14516}, {10117, 10984}, {10125, 15067}, {11424, 12302}, {11459, 14940}, {11695, 15059}, {11793, 12219}, {11801, 15026}, {12228, 13367}, {12358, 14448}, {13293, 15055}, {13348, 15036}, {13754, 14643}, {14094, 15012}, {15024, 15081}, {15028, 15100}, {15051, 15644}
= X(1568) of orthocentroidal triangle
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3, 11557, 13417), (5, 11561, 11562), (113, 14708, 185), (1986, 5972, 5562), (3567, 12383, 11800), (15024, 15102, 15081)
= [ 2.3153530443835720, 1.7206481696362790, 1.3808220362899220 ]
· Particular case 2: L = van Aubel line = X(4)X(6) of ABC
A*->A’ = X(51)X(2794) ∩ X(132)X(389)
= (SB+SC)*((4*R^2+3*SW)*S^4-(8*( -SW+3*SA)*R^4-2*(5*SW^2+5*SA^ 2-6*SB*SC)*R^2-(SA+SW)*(SA-3* SW)*SW)*S^2+(4*R^2-SW)*(4*R^2+ SA-2*SW)*SA*SW^2) : : (barys)
= X(52)+2*X(6720), X(112)+5*X(3567), X(127)-4*X(5462), X(132)+2*X(389), X(1297)-7*X(15043), 2*X(5446)+X(14689), 7*X(9781)-X(10735)
= on lines: {51, 2794}, {52, 6720}, {112, 3567}, {127, 5462}, {132, 389}, {1297, 15043}, {5446, 14689}, {9781, 10735}
= [ 1.0333183642805770, 0.9895936316878446, 2.4786450303017580 ]
A’->A* = X(51)X(2794) ∩ X(112)X(389)
= (SB+SC)*((7*R^2+SW)*S^4-(8*(3* SA-SW)*R^4-(10*SA^2-9*SB*SC+2* SW^2)*R^2+(SA+SW)*SW^2)*S^2+( 4*R^2-SW)^2*SA*SW^2) :: (barys)
= X(112)+2*X(389), 2*X(132)+X(185), X(1297)-4*X(9729), 5*X(3567)+X(13200), 4*X(5462)-X(10749), X(5562)-4*X(6720), 4*X(10110)-X(10735), 5*X(10574)+X(12384), X(13219)-7*X(15043)
= on lines: {51, 2794}, {112, 389}, {132, 185}, {1297, 9729}, {3567, 13200}, {5462, 10749}, {5562, 6720}, {10110, 10735}, {10574, 12384}, {13219, 15043}
= [ 1.9460111458536590, 2.1353474524540250, 1.2641880244299040 ]
· Note
For six different lines through H, it results:
1) Eulerologic centers A*->A’ lie on the circle with center X(5946) and radius R/6
2) Eulerologic centers A’->A* lie on the circle with center X(9730) and radius R/3
Both circles have these nice centers of similitude:
external = X(51)
internal = X(2)X(389) ∩ X(30)X(51)
= (SB+SC)*(4*S^2+SA*(SA+12*R^2- 3*SW)) : : (barys)
= X(2)+2*X(389), 4*X(2)-X(5562), 5*X(2)+X(5889), 11*X(2)-5*X(11444), 5*X(2)-8*X(11695), 7*X(2)-4*X(11793), 2*X(2)+X(14831), 7*X(2)-13*X(15028), X(2)-7*X(15043), 8*X(389)+X(5562), 10*X(389)-X(5889), 5*X(389)+4*X(11695), 7*X(389)+2*X(11793), 4*X(389)-X(14831), 2*X(389)+7*X(15043), 5*X(5562)+4*X(5889), 11*X(5562)-20*X(11444), 7*X(5562)-16*X(11793), X(5562)+2*X(14831), X(5889)+8*X(11695), 2*X(5889)-5*X(14831), 14*X(11695)-5*X(11793), 8*X(11793)+7*X(14831), 4*X(11793)-13*X(15028), X(14831)+14*X(15043)
= on lines: {2, 389}, {3, 15004}, {4, 15010}, {30, 51}, {52, 549}, {140, 14531}, {143, 8703}, {185, 381}, {186, 575}, {373, 5055}, {376, 3567}, {511, 3524}, {547, 6102}, {568, 3917}, {578, 15078}, {1092, 11432}, {1154, 5650}, {1216, 15694}, {1843, 11179}, {2071, 15019}, {2979, 15708}, {3060, 10304}, {3091, 13382}, {3534, 5446}, {3543, 10110}, {3545, 5890}, {3581, 15038}, {3796, 10245}, {3819, 15709}, {3839, 5640}, {3845, 11381}, {5066, 12162}, {5071, 5907}, {5422, 11438}, {5447, 15701}, {5476, 12294}, {5642, 9826}, {5655, 11806}, {5663, 14845}, {5876, 10109}, {5891, 13363}, {6101, 11812}, {6243, 15693}, {6644, 13366}, {6688, 11459}, {7527, 12834}, {7706, 13851}, {9781, 15682}, {9822, 11180}, {9909, 10984}, {10095, 10575}, {10303, 15606}, {10625, 12100}, {11001, 13598}, {11412, 15702}, {11430, 15053}, {11455, 13570}, {13321, 15688}, {13340, 15706}, {13348, 15698}, {13364, 16194}, {13491, 14893}, {14449, 14891}, {15644, 15692}
= midpoint of X(i) and X(j) for these {i,j}: {568, 5054}, {3060, 10304}, {3545, 5890}
= reflection of X(i) in X(j) for these (i,j): (3545, 5943), (5891, 15699)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2, 389, 14831), (2, 14831, 5562), (389, 11695, 5889), (568, 5892, 3917), (5890, 5943, 15030), (5946, 9730, 51)
= [ 2.8892883171792480, 2.6051987431920600, 0.5035476671533007 ]
and
Radical trace = X(30)X(51) ∩ X(389)X(468)
= (SB+SC)*((12*R^2+5*SW)*S^2-6* R^2*(12*R^2*SA+6*SB*SC-5*SA^2) +(SA-4*SW)*SA*SW) : : (barys)
= 2*X(389)+X(468), X(858)-7*X(15043), 2*X(974)+X(1514), 5*X(3567)+X(10295), 4*X(5462)-X(10297), 4*X(9826)-X(11064), 4*X(12006)-X(15122)
= on lines: {6, 186}, {30, 51}, {389, 468}, {403, 5890}, {597, 15045}, {858, 15043}, {974, 1514}, {1154, 9826}, {2071, 5422}, {3567, 10295}, {5462, 10297}, {5892, 10257}, {7464, 10982}, {7729, 11455}, {11695, 15739}, {12006, 15122}, {13352, 15646}
= midpoint of X(403) and X(5890)
= [ 2.4157103746488840, 2.1328701112317770, 1.0491188473705870 ]
César Lozada
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