Let ABC be a triangle and P a point.
Denote:
A', B', C' = the NPC centers of PBC, PCA, PAB, resp.
(Obc), (Ocb) = the circles (B, BC'), (C, CB'), resp.
(Oca), (Oac) = the circles (C, CA'), (A, AC'), resp.
(Oab), (Oba) = the circles (A, AB'), (B, BA'), resp.
Ra = the radical axis of (Obc), (Ocb)
Rb = the radical axis of (Oca), (Oac)
Rc = the radical axis of (Oab), (Oba)
Sa = the radical axis of (Oba), (Oca)
Sb = the radical axis of (Ocb), (Oab)
Sc = the radical axis of (Oac), (Obc)
Which is the locus of P siuch that:
1. Ra, Rb, Rc are concurrent?
2. Sa, Sb, Sc are concurrent?
The entire plane?
And which are the concurrence points in terms of P?
[César Lozada]:
In both cases, the locus is the entire plane.
1) For P=x:y:z (barys), Qr = Ra ∩ Rb ∩ Rc =
(-SB*c^6*y^4-SC*b^6*z^4-2*SA*SB*c^4*y^3*z-2*SA*SC*b^4*y*z^3-3*SA^3*SB*y^2*z^2-3*SA^3*SC*y^2*z^2+SA*y^2*z^2*(2*SA-3*SB-3*SC)*S^2+18*S^4*y^2*z^2)*x^4+2*(-SB^2*c^4*y^3-SC^2*b^4*z^3-3*SA^3*SC*y*z^2-3*SA*SB^3*y^2*z-3*SA*SC^3*y*z^2-3*SA^3*SB*y*z^2-3*SA^3*SC*y^2*z-3*SA^3*SB*y^2*z+y*z*(3*SA^2*z-3*SA*SB*z-4*SA*SC*z+SC^2*z-3*y*SA*SC-4*y*SA*SB+3*SA^2*y+SB^2*y)*S^2+14*y*z*(y+z)*S^4)*y*z*x^3+(-3*SA*(2*z*y*SC^3+SA^2*SB*y^2+8*SA^2*SB*y*z+SA^2*SB*z^2+SA^2*SC*y^2+8*SA^2*SC*y*z+SA^2*SC*z^2-SA*SB^2*y^2-4*SA*SB^2*y*z-4*SA*SC^2*y*z-SA*SC^2*z^2+y^2*SB^3+2*z*y*SB^3+z^2*SC^3)+(4*SW^2*y*z-6*SA*SB*y^2+3*y^2*SA^2+20*z*y*SA^2+3*z^2*SA^2+2*y^2*SB^2-3*z^2*SA*SB-3*y^2*SA*SC-6*SA*SC*z^2+22*z*y*SB*SC+2*z^2*SC^2)*S^2+(10*y^2+10*z^2-6*y*z)*S^4)*y^2*z^2*x^2+2*(SB*y+SC*z)*a^6*y^3*z^3*x+a^8*y^4*z^4 : :
On despite of this long expression, if P lies on de the circumcircle of ABC then Qr = centroid-of-quadrangle ABCP.
Examples: For P on the circumcircle of ABC, see ETC X(6666) to X(6723).
For P not lying on the circumcircle of ABC:
Qr(X(4)) = X(5)
Qr(X(2)) = X(2)X(1495) ∩ X(5)X(26614)
= 24*S^4+(3*SA-4*SW)*SW*S^2+9*SB*SC*SW^2 : : (barys)
= lies on these lines: {2, 1495}, {5, 26614}, {30, 1153}, {381, 5215}, {511, 7610}, {542, 9771}, {575, 6055}, {3830, 8588}, {6054, 10486}, {7603, 14830}, {13355, 21358}, {15597, 19924}, {23053, 31670}
= [ -5.6038809298123960, -7.4997772059989710, 11.4192245152047700 ]
Qr(X(4)) = X(2)X(32401) ∩ X(140)X(6000)
= (R^2*(188*R^2+SA-96*SW)+12*SW^2)*S^2-(R^2*(52*R^2-31*SW)+4*SW^2)*SB*SC : : (barys)
= 15*X(15694)+X(32321)
= lies on these lines: {2, 32401}, {140, 6000}, {2777, 5498}, {6143, 13289}, {6288, 11202}, {10018, 18383}, {10125, 18400}, {10182, 14076}, {13363, 15426}, {15694, 32321}, {16532, 32351}
= [ 2.5584839847759280, 1.5687514859004760, 1.3737669986951600 ]
Qr( X(6) ) = X(511)X(25488) ∩ X(3589)X(5066)
= 4*S^4-(3*R^2*(3*SA-4*SW)-4*SA^2+4*SB*SC+12*SW^2)*S^2-27*R^2*SB*SC*SW : : (barys)
= lies on these lines: {511, 25488}, {3589, 5066}, {3818, 5012}, {6676, 10219}, {7605, 32305}, {14389, 25561}
= [ -0.1211762981077650, -0.9443205235026893, 4.3503523665359780 ]
Qr( X(30) ) = X(3)X(31378) ∩ X(30)X(511)
= 5*S^4+3*(3*R^2*(12*R^2-3*SA-4*SW)+2*SA^2-SB*SC+SW^2)*S^2-3*(9*R^2*(12*R^2-5*SW)+5*SW^2)*SB*SC : : (barys)
= lies on these lines: {3, 31378}, {4, 5627}, {20, 14480}, {30, 511}, {74, 3258}, {107, 18809}, {113, 22104}, {133, 1552}, {146, 476}, {381, 18279}, {477, 1138}, {550, 18285}, {1304, 5667}, {3154, 20417}, {3233, 6053}, {5502, 20128}, {7471, 15063}, {7687, 12079}, {7728, 14993}, {10096, 16244}, {10620, 20957}, {10745, 16177}, {12041, 31379}, {13382, 14895}, {14508, 14731}, {14611, 16163}, {14934, 16111}, {15054, 17511}, {16978, 21649}
= [ 1.0716841958219100, 1.2418794074718680, -1.3543861417060210 ]
The last one has isogonal conjugate:
Qr( X(30) )* = X(107)X(12121) ∩ X(476)X(16163)
= (SB+SC)*(5*S^4+3*(3*R^2*(12*R^2-3*SB-4*SW)+2*SB^2-SA*SC+SW^2)*S^2-3*(9*R^2*(12*R^2-5*SW)+5*SW^2)*SC*SA)*(5*S^4+3*(3*R^2*(12*R^2-3*SC-4*SW)+2*SC^2-SA*SB+SW^2)*S^2-3*(9*R^2*(12*R^2-5*SW)+5*SW^2)*SA*SB) : : (barys)
= lies on the circumcircle and these lines: {107, 12121}, {476, 16163}, {1304, 1511}, {2132, 15035}
= [ 13.5998688195708300, 11.7360545569038300, -10.7611588972894200 ]
2) Qs = Sa ∩ Sb ∩ Sc = midpoint( X(3), P)
Some ETC pairs (P, Qs(P)): (1,1385), (2,549), (3,3), (4,5), (5,140), (6,182), (7,31657), (8,5690), (9,31658), (10,6684), (11,6713), (12,31659), (13,6771), (14,6774), (15,13350), (16,13349), (20,550), (21,5428), (22,7502), (23,7575), (25,6644), (26,1658), (30,30), (32,13335), (36,23961), (39,13334), (40,3579), (49,13367), (50,22463)
César Lozada
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