HYACINTHOS 29105

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

(Oa), (Ob), (Oc) = the circumcircles of PNbNc, PNcNa,PNaNb, resp.

Ra = the radical axis of (Ob), (Oc)
Rb = the radical axis of (Oc), (Oa)
Rc = the radical axis of (Oa), (Ob)

 

Which is the locus of P such that the reflections of Ra, Rb, Rc in AI, BI, CI, resp, are concurrent?
Part of the locus is the IN line.
As P moves on the IN line the point of concurrence moves on the OI line.

 

 

[César Lozada]: 

 

Locus = { Line IN }

Let Q(P) be the concurrence point. Then if P ∈ IN and IP=t*IN then Q lies on the line IO and IQ = -t/(2(t-1))*IO..

 

ETC pairs (P,Q(P)) for P ∈ IN: (1,1), (5,517), (11,65), (12,3057), (80,11009), (119,23340), (355,1482), (495,9957), (496,942), (952, 24680),  (1387,24928), (1484,6583), (1837,2099), (5219,1697), (5252,2098), (5443,35), (5587,7982), (5881,16200), (5886,3), (5901,1385), (7741,5903), (7951,5697), (7958,7957), (7988,7991), (7989,11531), (8227,40), (9578,7962), (9581,3340), (9624,3576), (10283,15178), (10826,25415), (10827,30323), (10886,12435), (10944,5048), (10948,5570), (10950,11011), (11373,999), (11374,3295), (11375,55), (11376,56), (15888,5919), (15950,2646), (16173,5563), (17718,3303), (17720,5710), (18357,11278), (19907,11567), (23708,46), (26470,24474)

 

Some others:

 

Q( X(1317) ) = X(1)X(3) ∩ X(11)X(3244)

= a*(-a+b+c)*(4*a^2-(b+c)*a-5*(b-c)^2) : : (barys)

= 5*X(1)-X(7280), 3*X(1)-X(21842)

= lies on these lines: {1, 3}, {11, 3244}, {145, 11376}, {210, 4861}, {497, 20057}, {519, 17606}, {946, 1317}, {1392, 3885}, {1483, 30384}, {1737, 17662}, {1836, 5734}, {1837, 3241}, {3585, 13274}, {3636, 5432}, {3689, 10912}, {3698, 9342}, {3881, 15558}, {3893, 4511}, {3962, 11260}, {4018, 30538}, {4342, 10543}, {4930, 7082}, {5252, 10595}, {5326, 15808}, {5433, 28234}, {5854, 27385}, {6284, 13607}, {7967, 12701}, {7972, 18480}, {10039, 10283}, {10589, 20050}, {10915, 25416}, {10944, 13464}, {12645, 23708}, {12672, 17660}, {12735, 12743}, {12758, 24475}, {14100, 15570}, {15174, 16142}

= X(7280)-of-Mandart-incircle triangle

= X(16868)-of-Ursa-minor triangle

= X(21844)-of-Hutson intouch triangle, when ABC is acute

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7982, 1388), (1388, 7982, 1155), (3333, 12703, 46)

= [ 14.8310940184617100, 12.5409986185521800, -11.8866856471494600 ]

 

Q( X(1411) ) = X(1)X(3) ∩ X(58)X(10703)

= a*(-a+b+c)*(a^5-(b^2-3*b*c+c^2)*a^3-2*(b-c)^2*a*b*c+(b^2-c^2)*(b-c)*a^2-(b^3+c^3)*(b-c)^2) : : (barys)

= lies on these lines: {1, 3}, {58, 10703}, {946, 1411}, {1201, 12740}, {1399, 2800}, {2361, 3878}, {2594, 21740}, {3073, 17638}, {3924, 11376}, {4514, 4861}, {5258, 24431}, {10950, 15955}, {11700, 18360}

= [ -2.9886694388881310, -2.1732458960229990, 6.5246052278717440 ]

 

Q( X(1421) ) = X(1)X(3) ∩ X(8)X(27528)

= a*(-a+b+c)*(a^5+a^3*b*c+(b+c)*a^4-(b^3-c^3)*(b-c)*a-(b^4-c^4)*(b-c)) : : (barys)

= lies on these lines: {1, 3}, {8, 27528}, {21, 16579}, {29, 4858}, {33, 5142}, {34, 990}, {58, 2906}, {84, 1061}, {191, 2361}, {201, 13329}, {938, 9538}, {946, 1421}, {950, 3100}, {1071, 2003}, {1203, 1858}, {1210, 6198}, {1386, 12711}, {1399, 1768}, {1473, 11396}, {1717, 3583}, {1718, 3585}, {1724, 24430}, {1727, 2964}, {1789, 4282}, {1829, 3220}, {1831, 5358}, {1854, 16466}, {1870, 4292}, {1891, 12610}, {2006, 6831}, {2654, 30117}, {3474, 4347}, {3577, 15176}, {3616, 20266}, {4276, 22347}, {4303, 8555}, {4306, 20277}, {4336, 18343}, {5125, 20268}, {5136, 20320}, {5218, 30142}, {5256, 10393}, {5722, 8144}, {5727, 7221}, {6986, 16577}, {7191, 12053}, {9785, 17024}, {10395, 26723}, {22072, 30115}, {28806, 30122}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 15803, 1060), (65, 9630, 1), (942, 18455, 1)

= [ -4.5483968059354320, -3.4611535559500220, 8.1361077772276730 ]

 

Q( X(1483) ) = X(1)X(3) ∩ X(4)X(20057)

= a*(4*a^3-5*(b+c)*a^2-2*(2*b-c)*(b-2*c)*a+5*(b^2-c^2)*(b-c)) : : (barys)

= 5*X(1)-X(3), 9*X(1)-X(40), 19*X(1)-3*X(165), 3*X(1)-X(1385), 3*X(1)+X(1482), 11*X(1)-3*X(3576), 7*X(1)-X(3579), 7*X(1)+X(7982), 21*X(1)-5*X(7987), 17*X(1)-X(7991), 11*X(1)+X(8148), 7*X(1)-3*X(10246), X(1)+3*X(10247), 13*X(1)+3*X(11224), 5*X(1)+X(11278), 15*X(1)+X(11531), 13*X(1)-X(12702), 4*X(1)-X(13624), 11*X(1)+5*X(16189), X(4)+7*X(20057)

= lies on these lines: {1, 3}, {4, 20057}, {5, 3244}, {8, 5067}, {10, 10283}, {30, 13607}, {140, 3636}, {145, 5056}, {355, 3241}, {392, 5645}, {515, 3853}, {518, 5097}, {519, 547}, {546, 28236}, {548, 28228}, {551, 5690}, {632, 15808}, {944, 3543}, {946, 1483}, {952, 3850}, {962, 3655}, {971, 15570}, {1056, 10525}, {1058, 10526}, {1125, 5844}, {1317, 12047}, {1320, 22935}, {1392, 14497}, {1656, 3632}, {1699, 18526}, {1902, 13596}, {2771, 25485}, {3083, 21551}, {3084, 21544}, {3090, 20050}, {3242, 5102}, {3533, 3616}, {3555, 5694}, {3622, 26446}, {3623, 3832}, {3626, 3628}, {3633, 5790}, {3652, 28461}, {3653, 15719}, {3654, 15708}, {3827, 15580}, {3881, 14988}, {3892, 24475}, {4297, 15686}, {4301, 28146}, {4701, 10172}, {4930, 6762}, {5059, 18481}, {5270, 10738}, {5288, 7489}, {5330, 16858}, {5734, 7967}, {5881, 18493}, {5882, 22791}, {6001, 26200}, {6684, 11812}, {6924, 25439}, {7486, 20054}, {7680, 32214}, {7681, 32213}, {7743, 10950}, {7984, 11699}, {8227, 12645}, {9856, 26088}, {10106, 12735}, {10179, 31837}, {10532, 18407}, {11237, 11928}, {11238, 11929}, {11522, 18525}, {12103, 28232}, {12245, 15702}, {12737, 17661}, {15690, 28194}, {15723, 25055}, {16126, 22936}, {16854, 19860}, {16864, 19861}, {18446, 31822}, {18483, 28224}

= midpoint of X(i) and X(j) for these {i,j}: {1, 24680}, {3, 11278}, {5, 3244}, {65, 10284}, {944, 22793}, {946, 1483}, {1319, 23960}, {1320, 22935}, {1385, 1482}, {3555, 5694}, {3579, 7982}, {5882, 22791}, {7984, 11699}, {11224, 17502}, {16126, 22936}

= reflection of X(i) in X(j) for these (i,j): (140, 3636), (3626, 3628), (5885, 5045), (9856, 26088), (9955, 13464), (9956, 5901), (13145, 13373), (13624, 15178), (15178, 1), (31663, 1385)

= (K798e)-complement of-X(8)

= (anti-Aquila)-complement of-X(5690)

= X(3853)-of-2nd circumperp triangle

= X(3861)-of-excenters-reflections triangle

= X(10226)-of-intouch triangle

= X(11250)-of-incircle-circles triangle

= X(12102)-of-excentral triangle

= X(15178)-of-5th mixtilinear triangle

= X(15331)-of-Hutson intouch triangle

= X(24680)-of-anti-Aquila triangle

= X(26086)-of-2nd anti-circumperp-tangential triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7982, 10246), (1482, 11531, 11278), (11531, 16200, 1482)

= [ 0.4172949132722050, 0.6391479051384405, 3.0055798183782870 ]

 

César Lozada