Denote:
A', B', C' = the reflections of A, B, C in L, resp.
A'h, B'h, C'h = the reflections of Ah, Bh, Ch in L, resp.
The circumcircles of PA'A'h, PB'B'h, PC'C'h are coaxial.
The axis is the PH line. Let P' be the other than P intersection.
Applications:
Let L be the Euler line.
1. Which is the locus of P' as P moves on a fixed line line? A line?
The OI line, for example?
2. Which is the locus of P' as P moves on a given circle? A circle?
If the circle is the circumcircle (or the NPC), then P' lies on the NPC (or on the circumcircle).
Which is the locus of P' as P moves on the incircle?
[César Lozada]:
In general, when P=u : v : w (trilinears), coordinates of P’ are:
> Applications:
> Let L be the Euler line.
P’ = b*c*(b^2*SB*v^2+c^2*SC*w^2-a* SA*u*(b*v+c*w))*SB*SC : : (trilinears)
P->P’ is a isoconjugation, except for Linf for which P’=X(4)
ETC pairs (P,P’): (1,1785), (2,468), (3,403), (5,186), (6,5523), (8,1878), (10,242), (11,108), (13,6110), (14,6111), (15,6116), (16,6117), (19,5179), (20,10151), (23,427), (24,2072), (25,858), (52,5962), (53,13509), (69,5140), (72,5146), (74,133), (80,1845), (98,132), (99,5139), (100,5521), (101,5190), (107,125), (108,11), (110,136), (111,1560), (112,115), (113,1300), (114,3563), (115,112), (118,917), (119,915), (120,15344), (122,1301), (125,107), (126,2374), (127,1289), (128,2383), (131,1299), (132,98), (133,74), (135,925), (136,110), (137,933), (146,12133), (147,12131), (148,5186), (149,1862), (150,5185), (153,12138), (155,16172), (186,5), (193,5203), (226,243), (235,2071), (242,10), (243,226), (264,5167), (265,1986), (281,8074), (297,1316), (316,1843), (378,11799), (381,10295), (389,6761), (393,15341), (403,3), (407,7424), (419,11007), (427,23), (428,5189), (429,1325), (430,5196), (442,2074), (458,5112), (468,2), (546,13619), (671,5095), (857,14119), (858,25), (860,3109), (868,7473), (915,119), (917,118), (925,135), (933,137), (935,5099),….
1)
Suppose P*=u : v : w (actual trilinears) is a point and that P moves on the tripolar of P*.
Then the locus of P’ is a circle centered at O* = midpoint of H and the polar-conjugate of P. Coordinates of O* are:
O* = b*c*SB*SC*(SA*a*u*(SB*v*b+SC* w*c)+(S^2+SB*SC)*b*c*v*w) : : (trilinears)
and squared radius:
ρ2 = a*b*c*(SA*SB*SC)2*(m-n)/p2
where
m = a*b*c*(u^2*v^2+u^2*w^2+v^2*w^ 2)
n = 2*u*v*w*(SA*a*u+SB*b*v+SC*c*w)
p = 2*S*(SA*SB*a*b*u*v+SA*SC*a*c* u*w+SB*SC*b*c*v*w)
Note that p=0 when P* lies on the conic SA*SB*a*b*u*v+SA*SC*a*c*u*w+ SB*SC*b*c*v*w=0, whose points have tripolars passing through H. Therefore, the circle degenerates to a line for all P on a line through H.
Also note that the expression m-n must be studied for determining complex radius. (Is always m-n>=0 ???)
ETC-pairs (P*, O*):
(2,4), (4,381), (10,15762), (25,6248), (76,1596), (92,946), (94,403), (107,525), (253,10002), (264,5480), (275,546), (278,355), (281,5805), (297,115), (318,7682), (321,15763), (324,3574), (331,7680), (343,6750), (393,1352), (427,6249), (459,3), (468,9880), (470,5478), (471,5479), (525,133), (648,523), (653,522), (685,2799), (850,132), (1585,6250), (1586,6251), (1897,514), (1972,6530), (2052,5), (2394,11251), (2501,114), (2592,1312), (2593,1313), (2986,10151), (2996,6623), (3580,136), (4232,7620), (5392,235), (5523,14672), (6330,1503), (6331,512), (6335,513), (6336,519), (7017,7681), (8791,13449), (8794,5907), (8796,3091), (11547,9927), (11794,14618), (13149,3900), (14165,10113), (14618,113), (14918,137), (15352,520), (15459,9033), (15466,6247), (16080,30), (16081,511), (16082,517)
Some others:
O* ( P on line IO ) = X(4)X(513) ∩ X(523)X(10151)
= (a^3-(b+c)^2*a+2*(b+c)*b*c)*( b-c)*(a^2-b^2+c^2)*(a^2+b^2-c^ 2) : : (barys)
= on lines: {4, 513}, {523, 10151}, {900, 7649}, {3064, 14321}, {4132, 14618}
= [ -11.8745610126524500, 4.3152335345349760, 6.1337617330689870 ]
O* ( P on Brocard axis ) = midpoint of X(4) and X(14618)
= (SA^2+SB*SC)*(SB-SC)*SB*SC : : (barys)
= on lines: {4, 512}, {523, 10151}, {804, 2489}, {924, 13851}, {1882, 7178}, {2501, 3566}, {2506, 5254}, {4108, 6995}, {4367, 5307}, {5523, 7651}
= midpoint of X(4) and X(14618)
= [ -9.3301428814176010, 1.7515395083665310, 6.7342799983079810 ]
O* ( P on Fermat axis ) = X(4)X(690) ∩ X(230)X(231)
= (SA^2-SB*SC)*(SB-SC)*SB*SC : : (barys)
= 3*X(1637)-2*X(6130)
= on lines: {4, 690}, {5, 6334}, {24, 14270}, {98, 3563}, {107, 110}, {114, 132}, {125, 136}, {230, 231}, {264, 14295}, {526, 1112}, {804, 12131}, {826, 14618}, {850, 6368}, {879, 6531}, {1177, 15328}, {2848, 9409}, {4232, 9185}, {5095, 9003}, {5466, 16080}
= polar conjugate of X(2966)
= Dao-Moses-Telv circle-inverse-of X(2501)
= polar circle-inverse-of X(11005)
= [ 12.3880244153903900, -10.9281551623697100, 5.4887606333678440 ]
O* ( P on Nagel line ) = midpoint of X(4) and X(7649)
= (a^3+(b+c)*a^2-(b^2+4*b*c+c^2) *a-(b+c)*(b^2-4*b*c+c^2))*(b- c)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : : (barys)
= on lines: {4, 2457}, {1598, 4057}, {2969, 5510}, {4786, 6994}
= midpoint of X(4) and X(7649)
= [ -3.6106388770179000, -1.7105430226985490, 6.4913352870147130 ]
César Lozada
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