Let ABC be a triangle and P a point on the Euler line such that PO/OH = t (number).
Denote:
(Na), (Nb), (Nc) = the NPCs of IBC, ICA, IAB, resp.
D = the Poncelet point of ABCI = Feuerbach point X(11)
The perpendicular to AI through D intersects again (Na), (Nb), (Nc) at Aa, Ab, Ac, resp.
The perpendicular to BI through D intersects again (Na), (Nb), (Nc) at Ba, Bb, Bc, resp.
The perpendicular to CI through D intersects again (Na), (Nb), (Nc) at Ca, Cb, Cc, resp.
Pa, Pb, Pc = same to P points of the triangles AaBaCa, AbBbCb, AcBcCc, resp.
(ie Pa lies on the Euler line of AaBaCa such that PaOa/OaHa = t [Oa, Ha = the O and H of AaBaCa , resp]. Similarly Pb, Pc])
Conjecture:
ABC, PaPbPc are orthologic.
The locus of the orthologic center (PaPbPc, ABC) is the OI line.
1. Which are the orthologic centers in terms of t?
2. Which is the locus of the other orthologic center (ABC, PaPbPc) as t varies?
Your conjecture is true and P can be any point on the Euler line!!!.
Assume OP/OH=t, not necessarily an invariant number.
The locus of the orthologic center (A,Pa)=Qa is the Feuerbach hyperbola=isogonal conjugate of the line IO. If Qa-1 is the isogonal conjugate of Qa then IQa-1/IO = -2*r/((t+1)*(R-2*r)), ie, Qa-1(t)= ((R-2*r)*t+R)*X(1)-2*r*X(3)
The locus of the orthologic center (Pa,A)=Qp is the line IO, with IQp/IO = (1-t)/2, ie, Qp(t) = (1+t)*X(1)+(1-t)*X(3)
ETC pairs (P,Qa(P)): (3,80), (4,1320), (5,13143), (20,104), (376,1156), (550,3065), (1012,3254), (1532,3680), (3149,12641), (6840,1389), (6905,8), (6906,11604), (6909,4)
ETC pairs (P,Qp(P)): (2,10246), (3,1385), (4,1), (5,15178), (20,3), (376,3576), (382,23339), (550,13624), (1657,3579), (3146,1482), (3529,40), (3534,17502), (3543,10247), (4190,16203), (5059,12702), (5073,11278), (6834,1388), (6868,10267), (6869,11249), (6872,16202), (6905,1319), (6906,2646), (6916,18443), (6934,56), (6935,13384), (6938,55), (6942,21842), (6948,10269), (7411,13151), (11001,165), (11541,11531), (15682,16200), (17538,7987), (17579,10202), (20420,5045)
Some others:
Qa(X(2)) = ISOGONAL CONJUGATE OF X(5126)
= a*(a^3-(4*b+c)*a^2-(b^2-6*b*c+c^2)*a+(b^2-c^2)*(4*b-c))*(a^3-(b+4*c)*a^2-(b^2-6*b*c+c^2)*a+(b^2-c^2)*(b-4*c)) : : (barys)
= 2*X(6594)-3*X(9623)
= lies on the Feuerbach hyperbola and these lines: {1, 6946}, {4, 12762}, {7, 952}, {8, 4767}, {9, 2802}, {11, 1000}, {21, 10914}, {100, 2320}, {104, 1155}, {517, 1156}, {519, 3254}, {1317, 18490}, {1320, 3935}, {1656, 7320}, {2800, 3062}, {2826, 23836}, {2829, 10307}, {3427, 12247}, {3577, 10698}, {3625, 6598}, {3887, 4792}, {4900, 12653}, {5424, 10087}, {5557, 10106}, {5559, 12053}, {5561, 9897}, {5854, 6601}, {6594, 9623}, {7319, 8148}, {7972, 14563}, {10039, 13606}, {11604, 12531}, {12641, 21630}, {12738, 17097}, {13602, 16173}
= midpoint of X(4900) and X(12653)
= reflection of X(7972) in X(14563)
= antigonal conjugate of X(1000)
= isogonal conjugate of X(5126)
= antipode of X(1000) in the Feuerbach hyperbola
= trilinear pole of the line {45, 650}
= [ 2.8950835864795370, 4.5917372232070790, -0.8744229433033853 ]
Qa(X(21)) = ANTIGONAL CONJUGATE OF X(943)
= (a^5-(b+c)*a^4-b*(2*b+c)*a^3+2*b*(b^2+c^2)*a^2+(b+c)*(b^3-c^3)*a-(b^2-c^2)^2*(b-c))*(a^5-(b+c)*a^4-c*(b+2*c)*a^3+2*c*(b^2+c^2)*a^2-(b+c)*(b^3-c^3)*a+(b^2-c^2)^2*(b-c)) : : (barys)
= lies on the Feuerbach hyperbola and these lines: {1, 6901}, {9, 21090}, {11, 943}, {21, 149}, {104, 5842}, {388, 15173}, {497, 15175}, {952, 17097}, {1156, 5762}, {1479, 3467}, {2320, 3434}, {2346, 6881}, {4302, 15446}, {5083, 5557}, {6596, 21630}, {7319, 10526}
= antigonal conjugate of X(943)
= isotomic conjugate of the anticomplement of X(17796)
= antipode of X(943) in the Feuerbach hyperbola
= [ 2.9832067233572690, 4.6347410860647900, -0.9448670653792151 ]
Qp(X(21)) = MIDPOINT OF X(1) AND X(10902)
= a*(2*a^6-3*(b+c)*a^5-(3*b^2-2*b*c+3*c^2)*a^4+6*(b^3+c^3)*a^3+4*b^2*c^2*a^2-3*(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*(b-c)^2) : : (barys)
= on lines: {1, 3}, {21, 912}, {78, 6883}, {140, 5440}, {225, 7510}, {226, 7491}, {284, 8609}, {355, 6861}, {382, 5715}, {443, 10806}, {515, 6841}, {631, 12649}, {938, 6954}, {944, 6824}, {950, 6842}, {952, 6675}, {971, 13743}, {1006, 14054}, {1100, 5755}, {1104, 5396}, {1125, 6881}, {1621, 21740}, {2320, 5768}, {3358, 12687}, {3487, 6868}, {3488, 6825}, {3526, 5705}, {3560, 18446}, {3616, 6826}, {3655, 5787}, {3683, 5694}, {3868, 6875}, {3897, 6857}, {3916, 7508}, {4313, 6850}, {4330, 16155}, {5178, 6989}, {5248, 5887}, {5259, 6326}, {5267, 12005}, {5436, 5720}, {5439, 6924}, {5603, 6869}, {5703, 6827}, {5722, 6863}, {5731, 6851}, {5735, 15696}, {5745, 13607}, {5761, 6987}, {5777, 7489}, {5780, 16857}, {5842, 11281}, {5901, 20420}, {6847, 10587}, {6882, 13411}, {6906, 13369}, {6928, 11374}, {6951, 11015}, {7497, 11363}, {8728, 10943}, {9942, 10179}, {10165, 10916}, {10526, 17718}, {11520, 21165}
= midpoint of X(1) and X(10902)
= X(5576)-of-2nd circumperp triangle
= X(7568)-of-hexyl triangle
= X(10902)-of-anti-Aquila triangle
= X(14130)-of-Ascella triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3576, 11249), (1, 7987, 12704), (1, 16208, 7982), (3, 1385, 13151), (1385, 15178, 1319), (3587, 7987, 3), (3601, 18443, 3), (6906, 18444, 13369), (10246, 16202, 1)
= [ 3.4996040080742170, 3.1842910996015130, -0.1790465907739988 ]
Qa(X(22)) = ANTIGONAL CONJUGATE OF X(1061)
= (a^8-c*a^7-2*b^2*a^6+c*(b-c)^2*a^5+2*c^2*(b^2+b*c-c^2)*a^4+c*(b^2+4*b*c+c^2)*(b-c)^2*a^3+2*(b^2-c^2)*b*(b^3-b*c^2+c^3)*a^2-(b^4-c^4)*(b^2-c^2)*c*a-(b^4-c^4)*(b^2-c^2)^2)*(a^8-b*a^7-2*c^2*a^6+b*(b-c)^2*a^5-2*b^2*(b^2-b*c-c^2)*a^4+b*(b^2+4*b*c+c^2)*(b-c)^2*a^3-2*(b^2-c^2)*c*(b^3-b^2*c+c^3)*a^2-(b^4-c^4)*(b^2-c^2)*b*a+(b^4-c^4)*(b^2-c^2)^2) : : (barys)
= lies on the Feuerbach hyperbola and on line {11, 1061}
= antigonal conjugate of X(1061)
= antipode of X(1061) in the Feuerbach hyperbola
= [ 3.3136066119333600, 2.2845164845349490, 0.5297192486447862 ]
Qp(X(22)) = MIDPOINT OF X(1) AND X(15177)
= a*(-a^2+b^2+c^2)*(2*a^7-(b+c)*a^6-2*(b^2-b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-2*(b^4+c^4)*a^3+(b^2-c^2)^2*(b+c)*a^2+2*(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c)) : : (barys)
= on lines: {1, 3}, {5, 11363}, {10, 7542}, {26, 1829}, {184, 912}, {355, 3549}, {515, 15760}, {518, 19131}, {944, 3547}, {946, 12605}, {952, 6676}, {1125, 11585}, {1386, 9967}, {1902, 7526}, {2072, 11230}, {3616, 6643}, {3817, 10297}, {5886, 18531}, {5901, 12362}, {6146, 12259}, {6639, 9956}, {6923, 11393}, {6928, 11392}, {7404, 7718}, {7494, 7967}, {7713, 9714}, {7723, 11699}, {7968, 10897}, {7969, 10898}, {9619, 23115}, {9715, 11396}, {9928, 19357}, {9955, 18404}, {10024, 18480}, {10165, 10257}, {10575, 12262}, {11720, 12358}, {12266, 12363}, {18563, 22793}, {18670, 22054}
= midpoint of X(1) and X(15177)
= X(15177)-of-anti-Aquila triangle
= [ -2.1283964802063450, -1.4628960255762120, 5.6357754904785320 ]
Qa(X(382)) = TRILINEAR POLE OF THE LINE X(650)X(15492)
= a*(2*a^3-(3*b+2*c)*a^2-(2*b^2-7*b*c+2*c^2)*a+(b^2-c^2)*(3*b-2*c))*(2*a^3-(2*b+3*c)*a^2-(2*b^2-7*b*c+2*c^2)*a+(b^2-c^2)*(2*b-3*c)) : : (barys)
= 2*X(15079)-3*X(16173)
= lies on the Feuerbach hyperbola and these lines: {4, 7972}, {8, 6702}, {79, 14217}, {80, 5048}, {84, 13253}, {952, 5560}, {1317, 5561}, {1392, 2802}, {1476, 11009}, {2098, 15446}, {2099, 15180}, {3065, 12737}, {5424, 5919}, {5557, 11011}, {7284, 16200}, {10308, 10698}, {12331, 21398}, {12740, 13143}
= trilinear pole of the line {650, 15492}
= [ 4.0161514190000940, 2.0883987378800490, 0.3412416238366093 ]
--------------------------------
A conjecture, which resulted to be true, was derived from the previous centers:
If P is a point on the Feuerbach hyperbola, then the antigonal conjugate of P is the reflection of P in the Feuerbach point X(11) (or the antipode of P w/r to the Feuerbach hyperbola).
César Lozada
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