ADGEOM 5028 * ADGEOM 5029

#5028

Dear friends, "Circumconics with asymptotes making a given angle" is the title of my recent article published in Forum Geometricorum

 

An update can be found at my personal blog. This update introduces a new center, the focus of a parabola:

 

Briefly, the locus of perspectors of circumconics whose asymptotes make a given angle is a conic. This conic becomes a parabola for some particular angle related to Brocard angle. The focus F of this parabola is the point I introduce. Some properties of this point are:

 

First barycentric coordinate: a^2*(2*a^6 - b^6 - c^6 - 3*a^4*(b^2 + c^2) - 6*a^2*(b^4 - 3*b^2*c^2 + c^4)) : :

 

ETC search numbers: {1.10017518584542021, 4.49336719756577036, 0.0220986439717287574}

 

On lines X(i)X(j) for these {i,j}: {{6,110},{126,3589},{141,6719},{182,14688},{187,1084},{511,14650},{518,11721},{524,5914},{543,597},{1296,5085},{1428,3325},{1503,5512},{1576,21309},{2330,6019},{2492,2780},{3618,14360},{5027,6088},{5050,11258},{5166,9019},{5480,23699},{6094,11166},{6096,21448},{10748,14561},{14654,14853},{14666,20423}}

 

Combos: {X[126]-2*X[3589], X[141]-2*X[6719], 2*X[182]-X[14688], X[1296]-3*X[5085], 5*X[3618]-X[14360], 3*X[5050]+X[11258], X[10748]-3*X[14561], X[14654]+3*X[14853], X[14666]+X[20423]}

 

Midpoint of X(i) and X(j) for these {i,j}: {{6,111},{14666,20423}}

 

Reflection of X(i) in X(j) for these {i,j}: {{126,3589},{141,6719},{14688,182}}

 

Best regards,

Francisco Javier Garcia Capitan

 

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#5029

Hi Francisco Javier,

 

Another property of this point: it is the radical trace of circles {{X(6),X(13),X(16)}} and {{X(6),X(14),X(15)}}.

 

Best regards,

Randy Hurson

HYACINTHOS 28716

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

Ha, Hb, Hc = the orthocenters of AB'C', BC'A', CA'B', resp.

Which is the locus of P such that 

 

1. A'B'C', HaHbHc are perspective?
I lies on the locus (Thanos Kalogerakis)

2. A'B'C', HaHbHc are orthologic?


[César Lozada]:

 

Let P=u:v:w (trilinears)

 

1)      The entire plane.

Perspector: Q(P) = u+cos(C)*v+cos(B)*w : :

ETC pairs (P,Q(P)): (1,942), (2,5943), (3,5), (4,389), (5,5462), (6,6), (15,11542), (16,11543), (19,9119), (20,5907), (22,21243), (25,13567), (26,5449), (30,13754), (32,5305), (40,5777), (50,16310), (52,13292), (54,12242), (55,226), (56,1210), (64,4), (66,9969), (68,12235), (69,14913), (74,7687), (84,5908), (109,15252), (110,5972), (113,9826),….

 

2)      The entire plane:

Zah(P) = A’->Ha = (c*v+b*w)*(-cos(A)*v*w+u*(u+v*cos(C)+w*cos(B))) : :

ETC pairs(P, Zah(P)): (1,65), (3,3), (4,52), (5,6153), (6,1992), (15,396), (16,395), (30,13754), (36,1319), (40,18239), (54,1493), (55,8545), (64,3146), (74,30), (98,511), (99,512), (100,513), (101,514), (102,515), (103,516), (104,517), (105,518), (106,519), (107,520), (108,521), (109,522), (110,523), (111,524), (112,525), (187,27088),….

 

Zha(P) = Ha->A’ = isogonal-conjugate-of-P

 

Particular cases:

Q(X(7)) = X(1)X(3688) ∩ X(7)X(2808)

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

= on lines: {1, 3688}, {7, 2808}, {77, 14520}, {511, 5728}, {674, 5572}, {916, 942}, {938, 5933}, {2389, 15587}, {2810, 18412}, {3211, 9306}, {3819, 11018}, {3917, 11020}, {4253, 20793}, {6738, 29311}, {7146, 21746}, {8680, 13563}, {9440, 20683}, {13754, 15939}, {17092, 22440}

= [ 1.2730105116285230, 1.4741497929728930, 2.0325559275592110 ]

 

Q(X(8)) = COMPLEMENT OF X(23154)

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

= 3*X(51)-X(3868), 3*X(51)-2*X(12109), 3*X(375)-2*X(3812), 2*X(942)-3*X(5943), 3*X(3819)-4*X(5044), 3*X(3819)-2*X(11573), X(3874)-3*X(15049), 5*X(3876)-3*X(3917), 5*X(5439)-6*X(6688), 3*X(10167)-4*X(17704), 3*X(10176)-X(23156), 3*X(10202)-4*X(11695), 2*X(13369)-3*X(16836)

= on lines: {1, 2810}, {2, 23154}, {51, 3868}, {63, 970}, {65, 23841}, {72, 511}, {78, 26892}, {181, 1046}, {185, 2808}, {329, 10441}, {355, 2818}, {375, 3812}, {386, 20805}, {389, 912}, {404, 3937}, {517, 12527}, {581, 20760}, {651, 1425}, {936, 3784}, {942, 5943}, {960, 8679}, {978, 1401}, {986, 23638}, {1331, 3145}, {1463, 24178}, {1757, 10822}, {1762, 7066}, {2390, 5836}, {2392, 3678}, {2842, 3754}, {3061, 23630}, {3157, 9306}, {3219, 22076}, {3732, 17499}, {3819, 5044}, {3869, 16980}, {3874, 15049}, {3876, 3917}, {3916, 15489}, {3927, 5752}, {3951, 26893}, {4339, 9309}, {4415, 18178}, {5396, 22458}, {5439, 6688}, {5462, 24475}, {5777, 5907}, {5904, 9052}, {6743, 29353}, {7078, 24320}, {7248, 11512}, {9021, 9969}, {9822, 24476}, {10110, 24474}, {10167, 17704}, {10176, 23156}, {10202, 11695}, {13369, 16836}, {13731, 21361}, {17114, 24174}, {17572, 26910}, {17768, 22300}

= midpoint of X(i) and X(j) for these {i,j}: {185, 12528}, {3869, 16980}

= reflection of X(i) in X(j) for these (i,j): (65, 23841), (3868, 12109), (5907, 5777), (11573, 5044), (24474, 10110), (24475, 5462), (24476, 9822)

= complement of X(23154)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51, 3868, 12109), (5044, 11573, 3819)

= [ 1.9049567398322360, -0.0271654519746181, 2.7802605302750070 ]

 

Zah( X(2) ) = X(49)X(575) ∩ X(51)X(524)

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

= X(6)-4*X(9822), X(6)+2*X(14913), X(69)+2*X(9969), 2*X(141)+X(1843), 4*X(141)-X(3313), 3*X(373)-2*X(597), 2*X(389)+X(15069), X(1205)-4*X(6698), 2*X(1352)+X(19161), 2*X(1843)+X(3313), X(1992)-3*X(5640), X(2979)-3*X(21356), 7*X(3090)-X(15073), 4*X(3589)-X(6467), 5*X(3618)+X(12272), 7*X(3619)-X(12220), 4*X(3628)-X(15074), 2*X(9822)+X(14913), X(9967)-4*X(24206), X(12162)-4*X(18553)

= on lines: {2, 2393}, {5, 5181}, {6, 1196}, {39, 1634}, {49, 575}, {51, 524}, {67, 3521}, {69, 3060}, {110, 12039}, {141, 427}, {159, 3796}, {338, 6248}, {373, 597}, {381, 511}, {384, 1632}, {389, 15069}, {542, 9730}, {895, 16042}, {1205, 6698}, {1235, 27373}, {1350, 1597}, {1352, 7706}, {1495, 19127}, {1568, 5480}, {1992, 5640}, {1995, 8542}, {2072, 9967}, {2386, 11286}, {2781, 15030}, {2882, 8370}, {2979, 21356}, {3003, 11328}, {3090, 15073}, {3564, 5946}, {3589, 6467}, {3618, 12272}, {3619, 12220}, {3628, 15074}, {3763, 9973}, {3818, 16194}, {3819, 21358}, {5085, 11202}, {5092, 12367}, {5169, 19510}, {5421, 20794}, {5476, 14845}, {5650, 8705}, {5651, 8541}, {5890, 11180}, {5892, 11179}, {6697, 26156}, {6776, 15045}, {7669, 13335}, {8547, 22112}, {10110, 11477}, {10151, 12294}, {10602, 11284}, {10984, 15581}, {11002, 11160}, {11451, 15531}, {11645, 14855}, {11898, 13321}, {12093, 17430}, {14810, 18859}, {15060, 18358}, {15533, 21849}, {16072, 23049}, {19126, 20987}, {21513, 22143}, {21969, 22165}, {22087, 23635}

= midpoint of X(i) and X(j) for these {i,j}: {2, 11188}, {69, 3060}, {599, 9971}, {1843, 3917}, {5890, 11180}, {5943, 14913}

= reflection of X(i) in X(j) for these (i,j): (6, 5943), (51, 16776), (3060, 9969), (3313, 3917), (3917, 141), (5891, 11178), (5943, 9822), (11179, 5892), (15060, 18358), (16194, 3818)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9306, 9813, 6), (9822, 14913, 6)

= [ 1.7570858801305440, 0.1073895966489724, 2.7553551241671470 ]

 

César Lozada

HYACINTHOS 26708

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

A', B', C' = the midpoints of AP, BP, CP, resp.

(Np), (Na), (Nb), (Nc) = the NPCs of A'B'C', PBC, PCA, PAB, resp.

Ra = the radical axis of (Np), (Na)
Rb = the radical axis of (Np), (Nb)
Rc = the radical axis of (Np), (Nc)

A*B*C* = the triangle bounded by Ra, Rb, Rc

For P = N:

1. ABC, A*B*C* are parallelogic.
Parallelogic centers?
[(ABC, A*B*C*) =: U lies on the circumcircle]

2. ABC, A*B*C* are orthologic.
Orthologic centers?
[(ABC, A*B*C*) = antipode of U in the circumcircle]


[Peter Moses]:

Hi Antreas,

1)
(ABC,  A*B*C*) = X(930).

(A*B*C*,  ABC) = 

a^14 b^2-5 a^12 b^4+11 a^10 b^6-15 a^8 b^8+15 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16+a^14 c^2-10 a^12 b^2 c^2+25 a^10 b^4 c^2-26 a^8 b^6 c^2+7 a^6 b^8 c^2+14 a^4 b^10 c^2-17 a^2 b^12 c^2+6 b^14 c^2-5 a^12 c^4+25 a^10 b^2 c^4-26 a^8 b^4 c^4+5 a^6 b^6 c^4-4 a^4 b^8 c^4+21 a^2 b^10 c^4-16 b^12 c^4+11 a^10 c^6-26 a^8 b^2 c^6+5 a^6 b^4 c^6+2 a^4 b^6 c^6-9 a^2 b^8 c^6+26 b^10 c^6-15 a^8 c^8+7 a^6 b^2 c^8-4 a^4 b^4 c^8-9 a^2 b^6 c^8-30 b^8 c^8+15 a^6 c^10+14 a^4 b^2 c^10+21 a^2 b^4 c^10+26 b^6 c^10-11 a^4 c^12-17 a^2 b^2 c^12-16 b^4 c^12+5 a^2 c^14+6 b^2 c^14-c^16 : : 

 

= lies on these lines: {5,195}

2)
(ABC, A*B*C*) = X(1141).

(A*B*C*,  ABC) =

= a^14 b^2-5 a^12 b^4+11 a^10 b^6-15 a^8 b^8+15 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16+a^14 c^2-10 a^12 b^2 c^2+17 a^10 b^4 c^2-2 a^8 b^6 c^2-17 a^6 b^8 c^2+22 a^4 b^10 c^2-17 a^2 b^12 c^2+6 b^14 c^2-5 a^12 c^4+17 a^10 b^2 c^4-10 a^8 b^4 c^4+5 a^6 b^6 c^4-12 a^4 b^8 c^4+21 a^2 b^10 c^4-16 b^12 c^4+11 a^10 c^6-2 a^8 b^2 c^6+5 a^6 b^4 c^6+2 a^4 b^6 c^6-9 a^2 b^8 c^6+26 b^10 c^6-15 a^8 c^8-17 a^6 b^2 c^8-12 a^4 b^4 c^8-9 a^2 b^6 c^8-30 b^8 c^8+15 a^6 c^10+22 a^4 b^2 c^10+21 a^2 b^4 c^10+26 b^6 c^10-11 a^4 c^12-17 a^2 b^2 c^12-16 b^4 c^12+5 a^2 c^14+6 b^2 c^14-c^16 : : 

 

= lies on these lines: {2,3}, {511,14140}, {5663,16337}, {13391,16336}

Best regards,
Peter Moses

HYACINTHOS 28698

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

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

Aa, Ab, Ac = the orthogonal projections of A on INa, INb, INc, resp.

The Euler line of AaAbAc passes through the Feuerbach point Fe.

Which point is Fe wrt triangle AaAbAc?


 [Randy Hutson]:

Hi Antreas,

Triangles AaAbAc and BcBbBc, CaCbCc (constructed cyclically) are all similar to the excentral triangle. While Fe lies on the Euler lines of AaAbAc, BcBbBc, and CaCbCc, it does not occupy the same position relative to each triangle, therefore it is not a center of any of them.  There are some interesting results, however:

The similitude center of BcBbBc and CaCbCc is the A-vertex of the intouch triangle, and cyclically ...

Let Sa be the AaAbAc-to-excentral similarity image of Fe, and define Sb, Sc cyclically.

Sa, Sb, Sc lie on the Euler line of the excentral triangle (line X(1)X(3)), and the centroid of SaSbSc is X(165) (centroid of excentral triangle).

The lines ASa, BSb, CSc concur in:

= ISOGONAL CONJUGATE OF X(1768)

Trilinears 1/(a^5 - a^4 (b + c) - a^3 (2 b^2 - 5 b c + 2 c^2) + 2 a^2 (b - c)^2 (b + c) + a (b - c)^2 (b^2 - b c + c^2) - b^5 + b^4 c + b c^4 - c^5) : :

= lies on these lines:  {484, 1785},  {516, 5080},  {517, 1456},  {522, 1768},  {910, 5537}, {1325, 5538}, {5536, 22464} et al

 

= isogonal conjugate of X(1768)
= [X(4)-Ceva conjugate of X(110)]-of-excentral triangle

Search = [25.675244349994029, 20.896520660086082, -22.676270290457076]

Best regards,
Randy Hutson

 

HYACINTHOS 28692

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

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

Fe = the Feuerbach point.

 

The NPCs of IFeNa, IFeNb, IFeNc are coaxial.

Second (other than the midpoint of IFe) intersection?

 

[Peter Moses]:

Hi Antreas,

= X(1)X(18341)∩X(11)X(11700)

2 a^10-3 a^9 b-5 a^8 b^2+9 a^7 b^3+2 a^6 b^4-9 a^5 b^5+4 a^4 b^6+3 a^3 b^7-4 a^2 b^8+b^10-3 a^9 c+14 a^8 b c-7 a^7 b^2 c-22 a^6 b^3 c+24 a^5 b^4 c+a^4 b^5 c-15 a^3 b^6 c+8 a^2 b^7 c+a b^8 c-b^9 c-5 a^8 c^2-7 a^7 b c^2+34 a^6 b^2 c^2-14 a^5 b^3 c^2-27 a^4 b^4 c^2+22 a^3 b^5 c^2+a^2 b^6 c^2-a b^7 c^2-3 b^8 c^2+9 a^7 c^3-22 a^6 b c^3-14 a^5 b^2 c^3+44 a^4 b^3 c^3-10 a^3 b^4 c^3-8 a^2 b^5 c^3-3 a b^6 c^3+4 b^7 c^3+2 a^6 c^4+24 a^5 b c^4-27 a^4 b^2 c^4-10 a^3 b^3 c^4+6 a^2 b^4 c^4+3 a b^5 c^4+2 b^6 c^4-9 a^5 c^5+a^4 b c^5+22 a^3 b^2 c^5-8 a^2 b^3 c^5+3 a b^4 c^5-6 b^5 c^5+4 a^4 c^6-15 a^3 b c^6+a^2 b^2 c^6-3 a b^3 c^6+2 b^4 c^6+3 a^3 c^7+8 a^2 b c^7-a b^2 c^7+4 b^3 c^7-4 a^2 c^8+a b c^8-3 b^2 c^8-b c^9+c^10 : : 

 

= lies on these lines: {1,18341}, {11,11700}, {109,16173}, {117,11715}, {942,1387}, {1125,3738}, {2802,6718}, {2817,6713}, {3576,10771}, {5450,11798}

Best regards,
Peter Moses.

HYACINTHOS 28686

[Antreas P. Hatzipolakis]:

 

 

 

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

 

Denote:

 

Ab, Ac = the orthogonal projections of A' on AC, AB, resp.

 

Abc, Acb = the orthogonal projections of Ab, Ac on AB, AC, resp.

 

A* = AbAbc ∩ AcAcb,

 

Similarly B*, C*.  

 

Which is the locus of P such that

 

1. ABC, A*B*C* are perspective?

I lies on the locus. Perspector = X(55)

 

2. ABC, A*B*C* are orthologic?

 

--------------------------------------------------------------------------------------------

 

 

[Ercole Suppa]

 

(1) locus locus of P such that ABC, A*B*C* are perspective = Darboux cubic K004  through X(i) for these i: {1,3,4,20,40,64,84,1490,1498,2130,2131,3182,3183,3345,3346,3347,3348,3353,3354,3355,3472,3473,3637}

 

*** pairs {P = X(i),Q = X(j)} for these {i,j}: {1,55},{3,6},{4,3},{20,25},{40,56},{64,154},{84,198},{1490,1436},{1498,64},{3182,7037},{3345,1035},{3346,1033}

 

*** some points Q = Q(X(i))

 

 

Q1 = Q(X(2130)) = ISOGONAL CONJUGATE OF X(14362)

 

= a^2 (3 a^4-(b^2-c^2)^2-2 a^2 (b^2+c^2)) (a^12-6 a^10 (b^2-c^2)-2 a^2 (b^2-c^2)^4 (3 b^2+5 c^2)+a^8 (15 b^4+14 b^2 c^2-29 c^4)+(b^2-c^2)^4 (b^4+10 b^2 c^2+5 c^4)-4 a^6 (5 b^6+5 b^4 c^2-b^2 c^4-9 c^6)+a^4 (15 b^8-20 b^6 c^2+50 b^4 c^4-36 b^2 c^6-9 c^8)) (a^12+6 a^10 (b^2-c^2)-2 a^2 (b^2-c^2)^4 (5 b^2+3 c^2)+(b^2-c^2)^4 (5 b^4+10 b^2 c^2+c^4)+a^8 (-29 b^4+14 b^2 c^2+15 c^4)+4 a^6 (9 b^6+b^4 c^2-5 b^2 c^4-5 c^6)+a^4 (-9 b^8-36 b^6 c^2+50 b^4 c^4-20 b^2 c^6+15 c^8)) : : (barys)

 

= lies on these lines:  {3,2130}, {154,1033}, {2060,14365}

 

=  isogonal conjugate of X(14362)

 

=ETC 6-9-13 search numbers [-0.0652824214609015903, -0.171691718261267074, 3.78965832753182627]

 

 

Q2 = Q(X(2131)) = X(3)X(2131) ∩ X(1436)X(7037)

 

= -a^2 (a^2-b^2-c^2) (a^8-4 a^6 (b^2-c^2)+(b^2-c^2)^4-4 a^2 (b^2-c^2) (b^2+c^2)^2+2 a^4 (3 b^4+2 b^2 c^2-5 c^4)) (a^8+4 a^6 (b^2-c^2)+(b^2-c^2)^4+4 a^2 (b^2-c^2) (b^2+c^2)^2+a^4 (-10 b^4+4 b^2 c^2+6 c^4)) (a^16-8 a^14 (b^2+c^2)-56 a^10 (b^2-c^2)^2 (b^2+c^2)-8 a^2 (b^2-c^2)^6 (b^2+c^2)+(b^2-c^2)^6 (b^4+14 b^2 c^2+c^4)+4 a^12 (7 b^4-10 b^2 c^2+7 c^4)+2 a^8 (b^2-c^2)^2 (35 b^4+114 b^2 c^2+35 c^4)-8 a^6 (b^2-c^2)^2 (7 b^6+25 b^4 c^2+25 b^2 c^4+7 c^6)+4 a^4 (b^2-c^2)^2 (7 b^8+50 b^4 c^4+7 c^8)) : : (barys) 

 

= lies on these lines: {3,2131}, {1436,7037} 

 

=ETC 6-9-13 search numbers [-4.35775772132550380, -6.90963960045842761, 10.4355339228366000]

 

 

Q3 = Q(X(3183)) = ISOGONAL CONJUGATE OF X(14361)

 

= a^2 (a^2-b^2-c^2) (a^8-4 a^6 (b^2-c^2)+(b^2-c^2)^4-4 a^2 (b^2-c^2) (b^2+c^2)^2+2 a^4 (3 b^4+2 b^2 c^2-5 c^4)) (a^8+4 a^6 (b^2-c^2)+(b^2-c^2)^4+4 a^2 (b^2-c^2) (b^2+c^2)^2+a^4 (-10 b^4+4 b^2 c^2+6 c^4)) : : (barys)

 

= S^4 + (96 R^4-SB SC-32 R^2 SW+2 SW^2) S^2 + 256 R^6 SB+256 R^6 SC-64 R^4 SB SC-128 R^4 SB SW-128 R^4 SC SW+24 R^2 SB SC SW+16 R^2 SB SW^2+16 R^2 SC SW^2-2 SB SC SW^2 : :

 

= lies on these lines: {3,1033}, {6,14092}, {154,577}, {198,1035}, {1032,3964}, {1092,15905}, {1105,20792},

{1598,13855}, {15341,16391} 

 

=  isogonal conjugate of X(14361)

 

=ETC 6-9-13 search numbers [1.82861473596034199, 2.00978721618410853, 1.40529730025983843]

 

 

Q4 = Q(X(3347)) = X(3)X(3341) ∩ X(6)X(2188)

 

= a^2 (a-b-c) (a^3+a^2 (b-c)-a (b-c)^2-(b-c) (b+c)^2) (a^3-a (b-c)^2+a^2 (-b+c)+(b-c) (b+c)^2) (a^9+3 a^8 (b+c)+4 a^2 b (b-c)^4 c (b+c)-(b-c)^6 (b+c)^3-6 a^5 (b^2-c^2)^2+8 a^3 (b^2-c^2)^2 (b^2+c^2)+a^6 (-8 b^3+4 b^2 c+4 b c^2-8 c^3)+2 a^4 (b-c)^2 (3 b^3-b^2 c-b c^2+3 c^3)-a (b^2-c^2)^2 (3 b^4+10 b^2 c^2+3 c^4)) : : (barys)

 

= lies on these lines: {3,3341}, {6,2188}, {25,1436}, {56,64} 

 

=ETC 6-9-13 search numbers [1.50993671784466044, 1.82772279812444519, 1.67842405958530454]

 

 

Q5 = Q(X(3348)) = ISOGONAL CONJUGATE OF X(14365)

 

= a^2 (a^4+b^4+2 b^2 c^2-3 c^4-2 a^2 (b^2-c^2)) (a^4-3 b^4+2 b^2 c^2+c^4+2 a^2 (b^2-c^2)) (5 a^12+(b^2-c^2)^6-10 a^10 (b^2+c^2)+36 a^6 (b^2-c^2)^2 (b^2+c^2)+a^8 (-9 b^4+34 b^2 c^2-9 c^4)-a^4 (b^2-c^2)^2 (29 b^4+54 b^2 c^2+29 c^4)+2 a^2 (b^2-c^2)^2 (3 b^6+13 b^4 c^2+13 b^2 c^4+3 c^6)) : : (barys)

 

= (16 R^2+SB+SC-4 SW) S^4 + (1536 R^6+64 R^4 SB+64 R^4 SC-16 R^2 SB SC-896 R^4 SW-16 R^2 SB SW-16 R^2 SC SW+4 SB SC SW+160 R^2 SW^2-8 SW^3) S^2 -1024 R^6 SB SC+640 R^4 SB SC SW-128 R^2 SB SC SW^2+8 SB SC SW^3 : :

 

= lies on these lines: {3,2130}, {6,14092}, {25,64}, {56,7037}, {1073,9786}, {1301,1498}, {1436,2155}, {1620,11589}, {2060,14362}, {15394,17928}

 

= isogonal conjugate of X(14365)

 

=ETC 6-9-13 search numbers [3.74187532712576789, 3.20124371999504683, -0.302600552147161575]

 

--------------------------------------------------------------------------------------------

 

(2) locus locus of P such that ABC, A*B*C* are orthologic = {Euler line} U {circumcircle} 

 

*** let W = orthologic center (ABC, A*B*C*)

 

*** pairs {P=X(i) ∈ Euler Line, W=X(j)} for these {i,j}: {2,69},{3,3},{4,68},{5,3519},{20,4},{21,72},{22,6},{23,895},{24,15316},{25,6391},{26,15317},{30,265},{186,5504},{376,4846},{401,290},{548,14861},{550,3521},{858,67},{1370,66},{1657,21400},{1658,16867},{1817,1439},{2071,74},{2475,18123},{2937,15002},{3146,15077},{3151,8044},{3522,15740},{3534,18550},{3843,14841},{4184,71},{4225,73},{4226,879},{4230,2435},{4236,10099},{5059,15749},{5189,18125},{6636,1176},{7391,18124},{7396,16774},{7471,14220},{7488,54},{7493,5486},{7560,1246},{8613,8795},{8703,13623},{10296,11564},{10298,3431},{10565,17040},{11413,64},{11414,3527},{11634,10097},{12225,6145},{15329,14380},{15331,16665},{15704,17505},{16049,65},{16386,11744},{18859,11559},{19772,2992},{19773,2993},{21312,3426}

 

*** pairs {P = X(i) ∈ circumcircle, W = X(j)} for these {i,j}: none

 

*** some points W = W(X(i)) :

 

 

W1 = W(X(27)) = X(3)X(307) ∩ X(4)X(916)

 

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

 

= lies on these lines: {3,307}, {4,916}, {6,226}, {64,516}, {66,674}, {71,440}, {73,6356}, {74,1305}, {272,1175}, {349,2893}, {912,1243}, {1246,15467}, {2218,7083}, {2772,11744}, {6817,8814}, {8804,21091}

 

= ETC 6-9-13 search numbers [0.928644861130306464, 4.12697450926022525, 0.354922962667158636]

 

 

W2 = W(X(28)) = X(3)X(6511) ∩ X(4)X(912)

 

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

 

= lies on these lines: {3,6511}, {4,912}, {6,169}, {54,10202}, {64,517}, {65,23604}, {66,518}, {69,20235}, {71,18674}, {72,21015}, {74,13397}, {81,1175}, {520,3657}, {1177,2836}, {1245,2650}, {2771,11744}, {3874,9028}, {3962,10693}, {8673,10099}, {9940,14528}

 

= ETC 6-9-13 search numbers [-1.01587836573791640, 6.26163956518048247, -0.225449817492301196]

 

 

W3 = W(X(29)) = X(4)X(5906) ∩ X(6)X(1210)

 

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

 

= lies on these lines: {4,5906}, {6,1210}, {64,515}, {66,8679}, {73,18641}, {2779,11744}

 

= ETC 6-9-13 search numbers [5.90756029059245174, 4.51279433805320889, -2.21014404009512785]

 

 

Best regards

Ercole Suppa

HYACINTHOS 28681

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.
For P = X(355) [ = the  midpoint of the orthocenter and the Nagel Point = the Fuhrmann circle center ] the triangles ABC, IOP share the same centroid G. (*)

Which is the locus of P such that the centroid of IOP lies on the Euler line of ABC?  

 
[Angel Montesdeoca]¨

*** The locus of P such that the centroid of IOP lies on the Euler line of ABC is the line through X(40) parallel to Euler line.

 The centroid of IOP is the projection of  P from X(1385) on a Euler Line.
 
 
 Pairs {P=X(i), Q=X(j)}, for {i, j}:  {30,30}, {40,3}, {355,2}, {3652,21}, {3654,3524}, {3679,5054}, {5690,549}, {5691,381}, {7330,16418}, {12438,26451}, {16139,21161}.
 
 Other pairs {P=X(i), Qi}:
 
 Q191 = a (3 a^6-3 a^5 (b+c)+a^4 (-6 b^2+b c-6 c^2)-2 b c (b^2-c^2)^2+6 a^3 (b^3+c^3)+a^2 (3 b^4+b^3 c+6 b^2 c^2+b c^3+3 c^4)-3 a (b^5-b^4 c-b c^4+c^5)) : :
 
 = (6 r+7 R) X[3] +  2 R X[4]

= lies on these lines: {1,22937}, {2,3}, {36,4870}, {79,5204}, {191,1385}, {399,16164}, {517,5426}, {551,22765}, {758,4930}, {993,3655}, {999,5427}, {1125,16159}, {1482,4428}, {2771,3576}, {3579,3922}, {3584,5172}, {3612,17637}, {3647,13465}, {3648,5303}, {3654,11849}, {3656,5248}, {3679,12331}, {3683,13624}, {5096,10168}, {5217,5441}, {5251,18524}, {5453,16948}, {7701,7987}, {10543,10573}, {11263,16150}, {12645,21677}, {15178,16126}, {16118,17605}, {16143,26202}, {18253,18526}, {25055,26286}

= the midpoint of X(21) and X(21161)
 
= reflection of X(i) in X(j), for these {i, j}: {3,21161}, {5055,15671}, {21161,5428}
 
 (6 - 9 - 13) - search numbers  of  Q191: (4.37889887160875, 3.49782730292207, -0.801938514704167).

=================================================

Q1709 = a (-3 a^6+3 a^5 (b+c)+4 b c (b^2-c^2)^2+a^4 (6 b^2-8 b c+6 c^2)-6 a^3 (b^3+c^3)+a^2 (-3 b^4+4 b^3 c-6 b^2 c^2+4 b c^3-3 c^4)+3 a (b^5-b^4 c-b c^4+c^5)) : :

= (3 r+R) X[3] +  2 R  X[4]

= lies on these liens: {2,3}, {35,18518}, {55,18519}, {84,24299}, {355,4421}, {498,18542}, {519,10679}, {551,5450}, {958,3654}, {999,11551}, {1001,3653}, {1319,7284}, {1385,1709}, {1470,3582}, {1482,11260}, {1727,2099}, {1836,18493}, {2077,19875}, {3058,10949}, {3085,18545}, {3189,12645}, {3241,12000}, {3295,22759}, {3652,12635}, {3655,4428}, {3656,10680}, {3679,11248}, {3829,11928}, {3873,10247}, {3928,24474}, {4302,18499}, {4640,12702}, {4653,18451}, {4861,8148}, {4870,22766}, {5010,18491}, {5204,9955}, {5217,18480}, {5432,18516}, {6001,10179}, {6284,18544}, {6767,12735}, {7171,13151}, {8069,11237}, {8071,11238}, {10039,18525}, {10056,10058}, {10269,25055}, {12686,24927}, {15171,18543}, {15338,18517}, {15446,26437}, {24467,24473}

=  midpoint of X(1012) and X(16370)

= reflection of X(i) in X(j), for these {i, j}: {3,16370}, {16370,6914}

 (6 - 9 - 13) - search numbers  of Q1709: (0.0180894385147809, -0.851478204778399, 4.22180042128568).
 
 ===========================================
 
Q1710 = a (-3 a^9+9 a^5 b (b-c)^2 c+3 a^8 (b+c)+2 b (b-c)^4 c (b+c)^3+a^7 (6 b^2-7 b c+6 c^2)+a^6 (-9 b^3+b^2 c+b c^2-9 c^3)+a (b^2-c^2)^2 (3 b^4-5 b^3 c+6 b^2 c^2-5 b c^3+3 c^4)-a^2 (b-c)^2 (3 b^5+3 b^4 c+10 b^3 c^2+10 b^2 c^3+3 b c^4+3 c^5)+a^4 (9 b^5-9 b^4 c+8 b^3 c^2+8 b^2 c^3-9 b c^4+9 c^5)+a^3 (-6 b^6+3 b^5 c+12 b^4 c^2-14 b^3 c^3+12 b^2 c^4+3 b c^5-6 c^6)) : : 

=  (9 r^2+20 r R+14 R^2-3 s^2) X[3] +   4 R (r+R)   X[4]

= lies on these lines: {2,3}, {1385,1710}, {2217,3655}
 
 (6 - 9 - 13) - search numbers  of Q1710: (2.16278810494452, 1.28756269241637,1 .75106503102942)
 
 ========================================
 
Q1762 = a (3 a^9-3 a^8 (b+c)-2 b (b-c)^4 c (b+c)^3+a^7 (-6 b^2+b c-6 c^2)+a^5 b c (3 b^2+16 b c+3 c^2)+a^6 (9 b^3+5 b^2 c+5 b c^2+9 c^3)-a (b^2-c^2)^2 (3 b^4-5 b^3 c+2 b^2 c^2-5 b c^3+3 c^4)+a^3 (b+c)^2 (6 b^4-21 b^3 c+22 b^2 c^2-21 b c^3+6 c^4)+a^2 (b-c)^2 (3 b^5+9 b^4 c+16 b^3 c^2+16 b^2 c^3+9 b c^4+3 c^5)-a^4 (9 b^5+3 b^4 c+8 b^3 c^2+8 b^2 c^3+3 b c^4+9 c^5)) : :

= (9 r^2+32 r R+28 R^2-3 s^2) X[3] +  4 R (r+2 R)  X[4]

= lies on these lines:  {2, 3},  {1385, 1762},  {11903, 12132}

 (6 - 9 - 13) - search numbers  of Q1762: (3.51501502392203, 2.63622240263833, 0.193272807501520).
 
 =================================================
 
Q2100 = a (4 a^4 b c-2 b c (b^2-c^2)^2-2 a^2 b c (b^2+c^2)-3 a^3 Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]+3 a (b^2+c^2) Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]) :  : 

=  (2R-3 OH)  X[3] - 2 R  X[4]

= lies on these lines: {2,3}, {182,15162}, {1385,2100}, {2575,15041}, {14500,20127}, {14810,15163}

 (6 - 9 - 13) - search numbers  of Q2100: (2.52150451259463, 1.64533279712562, 1.33781661654604).
 
 ===========================================
 
Q2101 = a (4 a^4 b c-2 b c (b^2-c^2)^2-2 a^2 b c (b^2+c^2)+3 a^3 Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]-3 a (b^2+c^2) Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]) : :

=  (3 OH+2R) X[3] - 2R X[4]

= lies on these lines: {2,3}, {182,15163}, {1385,2101}, {2574,15041}, {14499,20127}, {14810,15162}
 
 (6 - 9 - 13) - search numbers  of Q2101: (11.0432212757981, 10.1445690563372, -8.47937006900201).
 
 =========================================
 
Q2941 = a (-3 a^6-13 a^4 b c-3 a^5 (b+c)+2 b c (b^2-c^2)^2+3 a (b+c) (b^2+c^2)^2+a^2 (3 b^4+11 b^3 c+6 b^2 c^2+11 b c^3+3 c^4)) : : 

= (3 r^2+r R-3 s^2) X[3] +   2 r R X[4]

= lies on these lines: {2,3}, {1385,2941}

 (6 - 9 - 13) - search numbers  of Q2941: (7.35887107283956, 6.46993826224272, -4.23492557941730).
 
 ======================================
 
Q2960 = a (-3 a^9+7 a^6 b c (b+c)+2 b (b-c)^4 c (b+c)^3+12 a^5 b c (b^2-b c+c^2)+a^7 (6 b^2-7 b c+6 c^2)+a^2 b (b-c)^2 c (3 b^3-b^2 c-b c^2+3 c^3)-2 a^4 b c (6 b^3-b^2 c-b c^2+6 c^3)+a (b^2-c^2)^2 (3 b^4-2 b^3 c+6 b^2 c^2-2 b c^3+3 c^4)-a^3 (6 b^6+3 b^5 c-6 b^4 c^2+14 b^3 c^3-6 b^2 c^4+3 b c^5+6 c^6)) : : 

= (3 r^2+7 r R+7 R^2-3 s^2) X[3] +   2 R (r+R) X[4]

= lies on these lines: {2,3}, {1385,2960}

   (6 - 9 - 13) - search numbers  of Q2960: (-52.4799208109804, -53.2109973776780, 64.7005491945986).
 
 ========================================
 
Q3929 = a (-9 a^6+9 a^5 (b+c)+8 b c (b^2-c^2)^2+2 a^4 (9 b^2+b c+9 c^2)-18 a^3 (b^3+c^3)-a^2 (9 b^4+10 b^3 c+18 b^2 c^2+10 b c^3+9 c^4)+9 a (b^5-b^4 c-b c^4+c^5)) : : 

= (9 r+14 R) X[3] +  4 R X[4]

= lies on these lines: {2,3}, {527,3653}, {1385,3929}, {3655,5325}

 (6 - 9 - 13) - search numbers  of Q3929: (4.29693931414626, 3.41608395684161, -0.707519479358086).
 
 ========================================
 
Q6253 = -2 a^7+2 a^6 (b+c)-2 a^2 b (b-c)^2 c (b+c)+(b-c)^4 (b+c)^3-a (b-c)^2 (b+c)^4+4 a^3 b c (b^2-b c+c^2)+a^5 (3 b^2-2 b c+3 c^2)+a^4 (-3 b^3+b^2 c+b c^2-3 c^3) : : 

= (R-r) X[3]  - (r+2 R) X[4]

= lies on these lines: {2,3}, {11,18407}, {36,18406}, {56,18517}, {57,80}, {142,13151}, {355,529}, {386,13408}, {388,18518}, {495,18524}, {497,18499}, {515,5883}, {517,5891}, {519,24474}, {528,3656}, {542,4260}, {551,24299}, {553,18389}, {942,5434}, {952,3873}, {997,12699}, {1385,6253}, {1389,3241}, {1478,11502}, {1482,3189}, {1699,5840}, {1737,7354}, {2771,11246}, {3058,15950}, {3086,18544}, {3296,18526}, {3476,15934}, {3601,5443}, {3679,5709}, {3878,28194}, {4293,18519}, {4299,18761}, {4511,22791}, {5138,5476}, {5229,18542}, {5442,10483}, {5587,5841}, {5708,18391}, {5755,17330}, {5842,5886}, {6284,9955}, {6796,10197}, {7956,10738}, {7965,24466}, {9940,28208}, {9956,11827}, {10056,11501}, {10072,26475}, {10532,11239}, {10954,11237}, {11227,28160}, {11235,22753}, {11499,26332}, {11826,22793}, {12943,18516}, {14986,18543}, {15171,18493}

= midpoint of X(4) and X(17579)

= the reflection of X(i) in X(j), for these {i, j}: {7491,11113}, {11113,5}

 (6 - 9 - 13) - search numbers  of Q6253: (-2.47017667143147, -3.33318020589872, 7.08833231895956).
 
 =====================================
 
Q7701 = a (3 a^6-3 a^5 (b+c)+a^4 (-6 b^2+5 b c-6 c^2)-4 b c (b^2-c^2)^2+6 a^3 (b^3+c^3)+a^2 (3 b^4-b^3 c+6 b^2 c^2-b c^3+3 c^4)-3 a (b^5-b^4 c-b c^4+c^5)) : :

= (6 r+5 R) X[3] + 4 R X[4]

= lies on these lines: {1,13465}, {2,3}, {55,9897}, {191,1482}, {758,10247}, {993,3656}, {1001,18515}, {1385,7701}, {1621,12773}, {1749,2099}, {2771,5426}, {3647,8148}, {3653,5450}, {3655,5248}, {3679,11849}, {4265,11178}, {4421,5790}, {4428,22758}, {4995,10058}, {5204,16118}, {5901,14450}, {6690,10742}, {10543,12647}, {10902,28208}, {12409,25055}, {12702,22937}, {13089,26287}, {13624,16143}, {16132,26202}, {16159,18493}, {19875,26285}

= the reflection of X(i) in X(j), for these {i, j}: {5054,15672}, {10246,5426}

 (6 - 9 - 13) - search numbers  of Q7701: ( 1.97543484902115, 1.10070367911273, 1.96689969681965).
 
 =======================================
 
Q8141 = a (-3 a^9+2 a^6 b c (b+c)-2 a^2 b^2 (b-c)^2 c^2 (b+c)+b (b-c)^4 c (b+c)^3+a^5 b c (3 b^2-8 b c+3 c^2)+a^7 (6 b^2-2 b c+6 c^2)+a^4 b c (-3 b^3+b^2 c+b c^2-3 c^3)+a (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)+a^3 (-6 b^6+4 b^4 c^2-4 b^3 c^3+4 b^2 c^4-6 c^6)) : : 

= (3 r^2+11 r R+10 R^2-3 s^2) X[3] +   R (r+2 R) X[4]

= lies on these lines: {2,3}, {517,11202}, {1385,8141}, {3654,5285}

  (6 - 9 - 13) - search numbers  of Q8141: (0.173505684107798, -0.696471951165465, 4.04275782466456).
 
 =========================================
 
Q9572 = a (-6 a^9-22 a^5 b^2 c^2+a^6 b c (b+c)+2 a^4 b^2 c^2 (b+c)+2 b (b-c)^4 c (b+c)^3+a^7 (12 b^2-b c+12 c^2)-a^2 b (b-c)^2 c (3 b^3+7 b^2 c+7 b c^2+3 c^3)+2 a (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)-a^3 (b+c)^2 (12 b^4-27 b^3 c+28 b^2 c^2-27 b c^3+12 c^4)) : :

= (6 r^2+25 r R+26 R^2-6 s^2) X[3] +  2 R (r+2 R) X[4]

= lies on these lines: {2,3}, {1385,9572}.

 (6 - 9 - 13) - search numbers  of Q9572: (3.10192984221503, 2.22422695007094, 0.669155127759102).
 
 =========================================
 
Q9573 = a (6 a^9-7 a^6 b c (b+c)-2 b (b-c)^4 c (b+c)^3+a^7 (-12 b^2+7 b c-12 c^2)-2 a^5 b c (6 b^2-5 b c+6 c^2)-a^2 b (b-c)^2 c (3 b^3-b^2 c-b c^2+3 c^3)+2 a^4 b c (6 b^3-b^2 c-b c^2+6 c^3)-2 a (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)+a^3 (b+c)^2 (12 b^4-21 b^3 c+28 b^2 c^2-21 b c^3+12 c^4)) : : 

= (6 r^2+19 r R+14 R^2-6 s^2) X[3] +  2 R (r+2 R) X[4]

= lies on these lines: {2,3}, {1385,9573}, {10251,12645}

 (6 - 9 - 13) - search numbers  of Q9573 : (-25.5622765242673, -26.3643625162548, 33.6908892351303).
 
 ===========================================
 
Q10251 = -4 a^10+3 a^7 b c (b+c)-(b^2-c^2)^4 (b^2+c^2)+a^8 (9 b^2-3 b c+9 c^2)-6 a^5 b c (b^3+c^3)-2 a^6 (b^4-3 b^3 c+5 b^2 c^2-3 b c^3+c^4)+2 a^2 (b^2-c^2)^2 (3 b^4+4 b^2 c^2+3 c^4)-a^4 (b+c)^2 (8 b^4-13 b^3 c+16 b^2 c^2-13 b c^3+8 c^4)+3 a^3 b c (b^5-b^4 c-b c^4+c^5) : : 

=  (5 r^2+17 r R+14 R^2-5 s^2) X[3] + ((r+2 R)^2-s^2)  X[4]

= lies on these lines: {2,3}, {1385,10251}, {3656,18453}, {8251,25055}

 (6 - 9 - 13) - search numbers  of Q10251: (-3.38029401127337, -4.24089663321795, 8.13680554087682
 
 ========================================
 
Q11826 = -2 a^7+2 a^6 (b+c)-2 a^2 b (b-c)^2 c (b+c)-a (b-c)^4 (b+c)^2+(b-c)^4 (b+c)^3+4 a^3 b c (2 b^2-b c+2 c^2)+a^5 (3 b^2-10 b c+3 c^2)+a^4 (-3 b^3+b^2 c+b c^2-3 c^3) : :  

= (3 R-r) X[3] + r X[4]

= lies on these lines: {2,3}, {165,5841}, {517,5434}, {519,5884}, {528,3655}, {529,3654}, {553,24474}, {1385,3058}, {1768,3359}, {2077,3584}, {2550,18519}, {2829,26446}, {3474,12702}, {3576,5840}, {3579,7354}, {3820,10742}, {4299,22759}, {4304,13151}, {4316,7688}, {4413,18516}, {4430,5844}, {4995,26285}, {5298,15908}, {5790,14647}, {6284,7743}, {7080,18545}, {10056,11248}, {10072,10269}, {10106,16004}, {10246,15170}, {10247,11038}, {10270,19875}, {10310,11237}, {10385,16202}, {10525,11238}, {10950,13145}, {20292,22791}, {26200,28198}

= midpoint of X(i) and X(j), for these {i, j}: {376,17579}, {3058,11826}

= reflection of X(i) in X(j), for these {i, j}: {3058,1385}, {11113,549}, {24474,553}

 (6 - 9 - 13) - search numbers  of Q11826: (5.83581153198378, 4.95089659091247, -2.48033078810138).
 
 ===============================================

Q11827 = -2 a^7+2 a^6 (b+c)-2 a^2 b (b-c)^2 c (b+c)-a (b-c)^4 (b+c)^2+(b-c)^4 (b+c)^3-4 a^3 b c (b^2+b c+c^2)+a^5 (3 b^2+2 b c+3 c^2)+a^4 (-3 b^3+b^2 c+b c^2-3 c^3) : :

= (r+3 R) X[3] - r X[4]

= lies on these lines: {2,3}, {165,5840}, {226,13151}, {515,10176}, {517,3058}, {528,3654}, {529,3655}, {553,10202}, {580,3017}, {582,1834}, {912,17781}, {952,3681}, {997,18481}, {1385,5434}, {1482,15170}, {1708,5722}, {1737,3579}, {2550,18499}, {2551,18518}, {3336,16113}, {3428,11238}, {3475,10246}, {3576,5841}, {3582,11012}, {3583,7688}, {3584,10902}, {3586,3587}, {3820,18524}, {3925,18407}, {4302,11502}, {4654,18443}, {5178,5690}, {5298,26286}, {5506,5691}, {5584,10525}, {5758,15933}, {5842,26446}, {6253,9956}, {7354,13624}, {7742,10953}, {8148,15172}, {10056,10267}, {10072,11249}, {10157,28160}, {10268,19875}, {10385,10679}, {10526,11237}, {12702,15171}, {18544,19843}

= midpoint of X(i) and X(j), for these {i, j}: {376,11114}, {5434,11827}

= reflection of X(i) in X(j), for these {i, j}: {1482,15170}, {5434,1385}, {11112,549}

 (6 - 9 - 13) - search numbers  of Q11827: (7.72891425640891, 6.83900526255034, -4.66122266435458).
 
 ========================================
 
Q16113 = 5 a^7-5 a^6 (b+c)+a (b-c)^4 (b+c)^2-(b-c)^4 (b+c)^3+a^5 (-9 b^2+b c-9 c^2)-a^2 (b-c)^2 (3 b^3+b^2 c+b c^2+3 c^3)+a^4 (9 b^3-b^2 c-b c^2+9 c^3)+a^3 (3 b^4+b^3 c+10 b^2 c^2+b c^3+3 c^4) : : 

= (8 r+9 R) X[3]  - 2 r X[4]

= lies on these lines: {2,3}, {79,4870}, {519,16139}, {758,3655}, {1385,16113}, {2193,3163}, {3579,5441}, {3612,18977}, {3647,18481}, {3652,4297}, {3653,16159}, {3683,26202}, {5427,10072}, {5655,16164}, {8148,10385}, {10543,12702}, {16140,21578}, {18253,18525}, {22937,28204}

= the midpoint of X(i) and X(j), for these {i, j}: {376,15677}, {3534,13743}, {3651,15678}

= reflection of X(i) in X(j), for these {i, j}: {2,5428}, {381,15670}, {3651,8703}, {3830,6841}, {3845,10021}, {5499,12100}, {5655,16164}, {6175,549}, {6841,15673}, {13743,17525}, {15679,5499}

 (6 - 9 - 13) - search numbers  of Q16113: (7.33016096202258, 6.44130388942299, -4.20185096247273).

========================================

Q16138 = a (-3 a^6+3 a^5 (b+c)+5 b c (b^2-c^2)^2+a^4 (6 b^2-7 b c+6 c^2)-6 a^3 (b^3+c^3)+a^2 (-3 b^4+2 b^3 c-6 b^2 c^2+2 b c^3-3 c^4)+3 a (b^5-b^4 c-b c^4+c^5)),b (-3 a^4 b (b+c)-3 b (b-c)^3 (b+c)^2+a^5 (3 b+5 c)+6 a^2 (b^4-b^2 c^2)-2 a^3 (3 b^3-b^2 c+5 c^3)+a (3 b^5-7 b^4 c+2 b^2 c^3-3 b c^4+5 c^5)) : : 

= 2 (3 r+2 R) X[3] +  5 R X[4]

= lies on these lines: {2,3}, {104,551}, {191,11531}, {758,16200}, {944,4428}, {999,16133}, {1385,16138}, {1482,19919}, {1621,3655}, {2077,3828}, {2975,3656}, {3241,22758}, {3584,10058}, {3822,10728}, {4861,11278}, {5303,9955}, {5441,10039}, {5450,25055}, {5603,11194}, {10308,16132}, {11281,16116}

= midpoint of X(21161) and X(21669)

= reflection of X(i) in X(j), for these {i, j}: {3651,21161}, {21161,21}

 (6 - 9 - 13) - search numbers  of Q16138: (0.773702837727355, -0.0978581327919375, 3.35131880258156).
 
 =========================================
 
Q16309 = -10 a^10+4 a^9 (b+c)-(b-c)^4 (b+c)^6-2 a (b-c)^4 (b+c)^3 (b^2-b c+c^2)+a^8 (21 b^2-8 b c+21 c^2)-2 a^7 (5 b^3-b^2 c-b c^2+5 c^3)-2 a^6 (b^4-4 b^3 c+22 b^2 c^2-4 b c^3+c^4)+4 a^2 (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)+2 a^3 (b-c)^2 (b^5+3 b^4 c-2 b^3 c^2-2 b^2 c^3+3 b c^4+c^5)+2 a^5 (3 b^5-6 b^4 c+5 b^3 c^2+5 b^2 c^3-6 b c^4+3 c^5)+a^4 (-20 b^6+6 b^5 c+28 b^4 c^2-20 b^3 c^3+28 b^2 c^4+6 b c^5-20 c^6) : : 

= (6 r^2+21 r R+21 R^2-5 s^2) X[3] +  (3 r R+6 R^2-s^2) X[4]

= lies on these lines: {2,3}, {1385,16309}
 
 (6 - 9 - 13) - search numbers  of Q16309: (3.37240655223218, 2.49399013569761, 0.357560594625028).
 
 ============================================
 
Q19919 =  a (-6 a^6+6 a^5 (b+c)+5 b c (b^2-c^2)^2+4 a^4 (3 b^2-b c+3 c^2)-12 a^3 (b^3+c^3)-a^2 (6 b^4+b^3 c+12 b^2 c^2+b c^3+6 c^4)+6 a (b^5-b^4 c-b c^4+c^5)) : :

= (12 r+13 R) X[3] +  5 R X[4]

= lies on these lines: {2,3}, {1385,19919}, {1397,10222}, {5426,16200}, {7987,16138}, {11278,22937}, {11531,16139}
 
 (6 - 9 - 13) - search numbers  of Q19919: (3.77803286596185, 2.89854639696973, -0.109728961823213).
 
 ========================================
 
Q21375 = a (-3 a^6+2 b c (b^2-c^2)^2+2 a^2 b c (b^2+c^2)+a^4 (3 b^2-4 b c+3 c^2)-3 a^3 (b^3+c^3)+3 a (b^5+b^3 c^2+b^2 c^3+c^5)) : : 

= (9 r^2+14 r R-3 s^2) X[3] +  4 r R X[4]

= lies on these lines: {2,3}, {1385,21375}, {3052,12702}
 
 (6 - 9 - 13) - search numbers  of Q21375: (8.63812473977162, 7.74581722570067, -5.70865116962607).
 
 ================================================
 
Q21677 = -4 a^7+4 a^6 (b+c)-(b-c)^4 (b+c)^3+a (b-c)^2 (b+c)^4+a^5 (9 b^2+2 b c+9 c^2)+2 a^2 (b-c)^2 (3 b^3+4 b^2 c+4 b c^2+3 c^3)-a^4 (9 b^3+b^2 c+b c^2+9 c^3)-2 a^3 (3 b^4+2 b^3 c+4 b^2 c^2+2 b c^3+3 c^4) : : 

= (5 r+7 R) X[3] + (r+2 R) X[4]

= lies on these lines: {2,3}, {57,5444}, {119,17009}, {214,5745}, {519,24299}, {551,24474}, {758,10165}, {942,5298}, {997,18253}, {1385,21677}, {1737,10543}, {2771,11227}, {3601,5445}, {3649,22937}, {3655,5791}, {4260,10168}, {4995,24929}, {5427,5432}, {5708,16137}, {5709,25055}, {6174,12619}, {6699,16164}, {10246,24477}, {11231,21155}, {11281,16139}, {15174,18391}

= midpoint of X(i) and X(j), for these {i, j}: {2,21161}, {3524,15671}

 (6 - 9 - 13) - search numbers  of Q21677: (4.10499983769563, 3.22465082157627, -0.486401396581795).
 
 ===========================================
 
Q26921 = a (3 a^6-3 a^5 (b+c)-2 b c (b^2-c^2)^2-2 a^4 (3 b^2+b c+3 c^2)+6 a^3 (b^3+c^3)+a^2 (3 b^4+4 b^3 c+6 b^2 c^2+4 b c^3+3 c^4)-3 a (b^5-b^4 c-b c^4+c^5)) : :

= (3 r+5 R) X[3] + R X[4]

= lies on these lines: {2,3}, {36,4654}, {55,3654}, {56,3653}, {63,13151}, {517,4428}, {519,10267}, {527,10269}, {551,11249}, {582,19765}, {846,7986}, {912,3576}, {958,28204}, {971,17502}, {993,5325}, {997,13624}, {1385,11194}, {1480,8616}, {1708,24929}, {1737,5217}, {3189,5690}, {3241,16202}, {3428,3656}, {3601,10399}, {3679,10902}, {3715,12738}, {3828,6796}, {3873,10246}, {3927,4511}, {3928,18443}, {4995,8069}, {5010,11502}, {5248,28194}, {5260,18518}, {5298,8071}, {5434,7742}, {7330,7987}, {10072,26357}, {10202,21165}, {11012,25055}, {11496,28198}, {11499,19875}, {14831,22076}, {15931,22758}

= midpoint of X(3) and X(16418)

= reflection of X(3560) in X(16418)

 (6 - 9 - 13) - search numbers  of Q26921: (4.91829520915878, 4.03580069931406, -1.42333379107556).
 
Angel Montesdeoca

HYACINTHOS 28655

[Antreas P. Hatzipolakis]:

 
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

 

(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.

 

D = the Poncelet point of ABCP.

 

A", B", C" = the other than D intersections of PD and (Na), (Nb), (Nc), resp.

 

The circumcircles of PA'A", PB'B", PC'C" are coaxial.

 

Which is the other than D intersection in terms of P?

 

--------------------------------------------------------------------------------------------

 

 

[Ercole Suppa]

 

** If P=(x:y:z) (barys) then Q=(2 a^4 x-2 a^2 b^2 x-2 a^2 c^2 x+a^4 y-b^4 y+2 b^2 c^2 y-c^4 y+a^4 z-b^4 z+2 b^2 c^2 z-c^4 z) (a^2 x^2-b^2 x^2-c^2 x^2+a^2 x y-b^2 x y+a^2 x z-c^2 x z+a^2 y z) : : (barys)

 

** pairs {P=X(i),Q=X(j)} for these {i,j}: {1,11700},{2,5642},{3,1511},{5,10272},{8,1145},{20,16163},{21,16164},{22,16165},{23,1495},{24,20771},{25,20772},{26,20773},{76,5976},{140,13392},{146,1553},{147,6072},{148,6071},{149,6075},{150,14505},{152,6074},{153,6073},{316,325},{382,1539},{549,11694},{671,16092},{858,11064},{962,1537},{1916,16068},{3146,13202},{3153,1568},{3448,6070},{3524,11693},{3557,2029},{3558,2028},{5011,910},{5057,908},{5080,17757},{5134,17747},{5176,6735},{5196,18653},{5523,16318},{6785,6784},{6787,6786},{6788,3756},{6792,6791},{6794,6793},{7464,10564},{7471,3233},{10152,1552},{10296,1531},{10989,13857},{11050,15354},{12084,25487},{12699,12611},{12833,15631},{13137,15630},{13509,8779},{14262,10354},{14360,6077},{14807,14499},{14808,14500},{15342,14999},{17511,3258},{18328,1146},{18339,2968},{18343,4904},{20344,14506},{21290,14507}

 

** some points:

 

Q(X(6)) =  MIDPOINT OF X(6) AND X(112)

 

= a^2 (a^4-b^4+b^2 c^2-c^4) (2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6) :: (barys) 

 

= X[127]-2*X[3589], X[141]-2*X[6720], X[1297]-3*X[5085], 5*X[3618]-X[13219], 3*X[5050]+X[13310], X[10749]-3*X[14561], 5*X[12017]-X[13115], X[12384]+3*X[25406], X[13200]+3*X[14853], X[13221]+3*X[16475], 3*X[16225]-X[19161], 2*X[19130]-X[19163]

 

= lies on these lines: {6,74}, {127,3589}, {132,1503}, {141,6720}, {518,11722}, {611,13312}, {613,13311}, {1297,5085}, {1384,14649}, {1428,3320}, {1691,13195}, {1974,13166}, {2330,6020}, {2492,6593}, {2794,5480}, {2799,5026}, {3618,13219}, {5039,14676}, {5050,13310}, {8744,18374}, {9019,10317}, {9142,21309}, {9157,17810}, {10749,14561}, {11610,14495}, {12017,13115}, {12145,19124}, {12384,25406}, {13200,14853}, {13221,16475}, {16225,19161}, {19130,19163}

 

= midpoint of X(6) and X(112)

 

= reflection of X(i) in X(j) for these {i,j}: {127,3589}, {141,6720}, {19163,19130}

 

= (6-8-13) search numbers [0.396222886788812052, 0.916360699355107193, 2.82338881922062241]

 

 

Q(X(7)) =  MIDPOINT OF X(7) AND X(934)

 

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

 

= 2*X[142]-X[5514], X[972]-3*X[21151]

 

= lies on these lines: {7,104}, {142,5514}, {658,13257}, {971,1543}, {972,21151}, {1360,3323}, {3321,12831}, {4617,15252}, {6366,10427}

 

= midpoint of X(7) and X(934)

 

= reflection of X(5514) in X(142)

 

= (6-8-13) search numbers [0.393551197518337608, 0.365477276887734759, 3.20600273751517614]

 

 

Q(X(9)) =  MIDPOINT OF X(9) AND X(101)

 

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

 

= X[103]-3*X[21153], X[116]-2*X[6666], X[142]-2*X[6710], X[150]-5*X[18230]

 

= lies on these lines: {2,14154}, {9,48}, {103,21153}, {116,6666}, {118,516}, {142,6710}, {150,18230}, {518,11712}, {528,21090}, {954,11028}, {1001,2809}, {3022,15837}, {3887,6594}, {5375,16586}, {5526,15730}

 

= midpoint of X(9) and X(101)

 

= reflection of X(i) in X(j) for these {i,j}: {116,6666}, {142,6710}

 

= (6-8-13) search numbers [2.44623971779084346, 0.550310010072905776, 2.13064691287659351]

 

 

Q(X(10)) = MIDPOINT OF X(10) AND X(101)

 

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

 

= 3*X[2]+X[1282], X[103]-3*X[10164], X[116]-2*X[3634], X[150]-5*X[1698], X[152]+3*X[165], 3*X[551]-X[10695], 7*X[9780]+X[20096], 3*X[10175]-X[10739]

 

= lies on these lines: {2,1282}, {10,98}, {103,10164}, {116,3634}, {118,516}, {120,24685}, {150,1698}, {152,165}, {519,11712}, {544,3828}, {551,10695}, {1125,2809}, {1362,3911}, {2786,9508}, {2801,3035}, {2808,6684}, {2810,6686}, {3033,6685}, {3842,6690}, {4712,24582}, {6541,17927}, {9780,20096}, {10175,10739}, {11028,13405}, {13411,18413}, {14543,21914}

 

= midpoint of X(10) and X(101)}

 

= reflection of X(i) in X(j) for these {i,j}: {116,3634}, {1125,6710}

 

= (6-8-13) search numbers [2.20970283517814542, 0.326175810443645090, 2.39498761228732692]

 

 

Q(X(11)) =  MIDPOINT OF X(11) AND (2720)

 

= (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+5 a^3 b c-2 a^2 b^2 c-3 a b^3 c+b^4 c-2 a^3 c^2-2 a^2 b c^2+4 a b^2 c^2+2 a^2 c^3-3 a b c^3+a c^4+b c^4-c^5) (2 a^7-2 a^6 b-3 a^5 b^2+3 a^4 b^3+a b^6-b^7-2 a^6 c+8 a^5 b c-3 a^4 b^2 c-4 a^3 b^3 c+4 a^2 b^4 c-4 a b^5 c+b^6 c-3 a^5 c^2-3 a^4 b c^2+8 a^3 b^2 c^2-4 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+3 a^4 c^3-4 a^3 b c^3-4 a^2 b^2 c^3+8 a b^3 c^3-3 b^4 c^3+4 a^2 b c^4-a b^2 c^4-3 b^3 c^4-4 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7) :: (barys)

 

= X[11]+X[2720], X[1737]+X[15524], X[2745]-3*X[21154]

 

= lies on these lines: {11,2720}, {521,3035}, {522,10271}, {1737,15524}, {2745,21154}, {3660,6001}

 

= midpoint of X(i) and X(j) for these {i,j}: {11,2720}, {1737,15524}

 

= (6-8-13) search numbers [1.24261794046012197, 1.81750078681728600, 1.80887873389850942]

 

 

Best regards

Ercole Suppa

 

HYACINTHOS 28651

[Tran Quang Hung]:

 

 

 

Let ABC be a triangle with NPC center N.

 

The reflection of the circle (NBC) in the lines CA, AB meets BC at Cb, Bc, resp.

 

Define similarly the points Ac, Ca and Ba, Ab.

 

Let Oa, Ob, Oc be the circumenters of the triangles AAbAc, BBcBa, CCaAb, resp.

 

Then X(186) of the triangle OaObOc lies on the Euler of ABC. Which is this point?

 

Let Ka, Kb, Kc be the circumcenters of the triangles ABaCa, BCbAb, CAcBc, resp.

 

Then X(186) of the triangle KaKbKc lies on the Euler line of ABC. Which is this point?

 

Also, if H1 and H2 be the orthocenters of the triangles OaObOc and KaKbKc, resp. then the line H1H2 is parallel to the Euler line of ABC. Which is this line?

 

--------------------------------------------------------------------------------------------

 

 

[Ercole Suppa]

 

Dear Tran Quang Hung,

 

X(186) of the triangle OaObOc is the point:

 

P1 = (name pending)

 

= 2 a^22-9 a^20 (b^2+c^2)+4 a^18 (3 b^4+8 b^2 c^2+3 c^4)-(b^2-c^2)^8 (b^6+b^4 c^2+b^2 c^4+c^6)+a^16 (3 b^6-41 b^4 c^2-41 b^2 c^4+3 c^6)+a^14 (-20 b^8+28 b^6 c^2+74 b^4 c^4+28 b^2 c^6-20 c^8)+2 a^2 (b^2-c^2)^6 (2 b^8-b^6 c^2-b^2 c^6+2 c^8)+2 a^10 b^2 c^2 (11 b^8+3 b^6 c^2+26 b^4 c^4+3 b^2 c^6+11 c^8)-2 a^8 (b^2-c^2)^2 (b^10+3 b^8 c^2-6 b^6 c^4-6 b^4 c^6+3 b^2 c^8+c^10)-a^4 (b^2-c^2)^4 (5 b^10-17 b^8 c^2-b^6 c^4-b^4 c^6-17 b^2 c^8+5 c^10)+a^12 (14 b^10-24 b^8 c^2-71 b^6 c^4-71 b^4 c^6-24 b^2 c^8+14 c^10)+2 a^6 (b^2-c^2)^2 (b^12-10 b^10 c^2+b^8 c^4-6 b^6 c^6+b^4 c^8-10 b^2 c^10+c^12) : : (barys)

 

= R^2 S^4 + (-92 R^6-21 R^2 SB SC+99 R^4 SW+4 SB SC SW-35 R^2 SW^2+4 SW^3) S^2 + 132 R^6 SB SC-157 R^4 SB SC SW+63 R^2 SB SC SW^2-8 SB SC SW^3 : : (barys)

As a point on the Euler line, X() has Shinagawa coefficients {92 R^6-99 R^4 SW-4 SW^3-R^2 (S^2-35 SW^2), -132 R^6+157 R^4 SW-4 S^2 SW+8 SW^3+21 R^2 (S^2-3 SW^2)}

 

= lies on this line: {2,3}

 

= (6-8-13) search numbers [-10.7997916222310703, -11.6408214221380734, 16.6842139074943085]

 

***************

 

X(186) of the triangle KaKbKc is the point:

 

P2 = (name pending)

 

= 2 a^22-11 a^20 (b^2+c^2)+23 a^18 (b^2+c^2)^2-(b^2-c^2)^8 (b^6+c^6)-a^16 (19 b^6+67 b^4 c^2+67 b^2 c^4+19 c^6)+a^14 (-6 b^8+28 b^6 c^2+62 b^4 c^4+28 b^2 c^6-6 c^8)+a^2 (b^2-c^2)^6 (5 b^8-3 b^6 c^2-5 b^4 c^4-3 b^2 c^6+5 c^8)-a^4 (b^2-c^2)^4 (9 b^10-11 b^8 c^2-7 b^6 c^4-7 b^4 c^6-11 b^2 c^8+9 c^10)+a^8 (b^2-c^2)^2 (12 b^10+13 b^8 c^2+29 b^6 c^4+29 b^4 c^6+13 b^2 c^8+12 c^10)+a^12 (28 b^10+18 b^8 c^2-13 b^6 c^4-13 b^4 c^6+18 b^2 c^8+28 c^10)+2 a^6 (b^2-c^2)^2 (2 b^12-5 b^10 c^2-6 b^6 c^6-5 b^2 c^10+2 c^12)-a^10 (28 b^12+7 b^10 c^2+13 b^8 c^4-24 b^6 c^6+13 b^4 c^8+7 b^2 c^10+28 c^12) : : (barys)

 

= (5 R^2-2 SW) S^4 + (-160 R^6-51 R^2 SB SC+164 R^4 SW+14 SB SC SW-55 R^2 SW^2+6 SW^3) S^2 + 192 R^6 SB SC - 212 R^4 SB SC SW+81 R^2 SB SC SW^2-10 SB SC SW^3 : : (barys)

 

As a point on the Euler line, X() has Shinagawa coefficients {(5 R^2 - 2 SW) (32 R^4 - S^2 - 20 R^2 SW + 3 SW^2), -192 R^6 + 212 R^4 SW - 14 S^2 SW + 10 SW^3 + R^2 (51 S^2 - 81 SW^2)}

 

= lies on this line: {2,3}

= (6-8-13) search numbers [-7.19335336272051906, -8.04389703781748783, 12.5295255216521102]

 

*************** 

 

The line H1H2 is the trilinear polar of the point

 

P3 = (a-b) (a+b) (a-c) (a+c) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^16 (b^2-c^2)+b^2 (b^2-c^2)^6 (b^2+c^2)^2-a^14 (4 b^4+b^2 c^2-5 c^4)+a^12 (4 b^6+10 b^4 c^2-b^2 c^4-9 c^6)-a^2 (b^2-c^2)^4 (4 b^8+6 b^6 c^2+4 b^4 c^4+5 b^2 c^6+c^8)+a^10 (4 b^8-16 b^6 c^2-14 b^4 c^4-3 b^2 c^6+5 c^8)+a^8 (-10 b^10+15 b^8 c^2+10 b^6 c^4+8 b^4 c^6+8 b^2 c^8+5 c^10)+a^4 (b^2-c^2)^2 (4 b^10+4 b^8 c^2-b^6 c^4-b^4 c^6+9 b^2 c^8+5 c^10)+a^6 (4 b^12-9 b^10 c^2+5 b^8 c^4+4 b^6 c^6+8 b^4 c^8-3 b^2 c^10-9 c^12)) (a^16 (b^2-c^2)-c^2 (b^2-c^2)^6 (b^2+c^2)^2+a^14 (-5 b^4+b^2 c^2+4 c^4)+a^12 (9 b^6+b^4 c^2-10 b^2 c^4-4 c^6)+a^10 (-5 b^8+3 b^6 c^2+14 b^4 c^4+16 b^2 c^6-4 c^8)+a^2 (b^2-c^2)^4 (b^8+5 b^6 c^2+4 b^4 c^4+6 b^2 c^6+4 c^8)-a^8 (5 b^10+8 b^8 c^2+8 b^6 c^4+10 b^4 c^6+15 b^2 c^8-10 c^10)-a^4 (b^2-c^2)^2 (5 b^10+9 b^8 c^2-b^6 c^4-b^4 c^6+4 b^2 c^8+4 c^10)+a^6 (9 b^12+3 b^10 c^2-8 b^8 c^4-4 b^6 c^6-5 b^4 c^8+9 b^2 c^10-4 c^12)) : : (barys)

 

= (6-8-13) search numbers [1.08939964531861435, 3.18120003404388610, 0.935495391268482101]

 

 

Best regards

Ercole Suppa

HYACINTHOS 28651

[Tran Quang Hung]:

 

Let ABC be a triangle with NPC center N.

The reflection of the circle (NBC) in the line BC meets CA, AB again at Ac, Ab, resp.

Define similarly the points Ba, Bc, Cb, Ca.

Let Oa, Ob, Oc be the circumcenters of the triangles AAcAb, BBaBc, CCbCa, resp.

Then the NPC center of the triangle OaObOc lies on the Euler line of ABC.
Which is this point?

Let Ka, Kb, Kc be the circumcenters of the triangles ACaBa, BCbAb, CAcBc, resp.

Then the orthocenter of the triangle KaKbKc lies on the Euler line of ABC.
Which is this point?


[Peter Moses]:

Hi Antreas,

> Then the NPC center of the triangle OaObOc lies on the Euler line of ABC. 

Which is this point?  

X(20030)

> Then the orthocenter of the triangle KaKbKc lies on the Euler line of ABC. Which is this point?

2 a^16-11 a^14 b^2+25 a^12 b^4-29 a^10 b^6+15 a^8 b^8+3 a^6 b^10-9 a^4 b^12+5 a^2 b^14-b^16-11 a^14 c^2+34 a^12 b^2 c^2-35 a^10 b^4 c^2+12 a^8 b^6 c^2-9 a^6 b^8 c^2+26 a^4 b^10 c^2-25 a^2 b^12 c^2+8 b^14 c^2+25 a^12 c^4-35 a^10 b^2 c^4+12 a^8 b^4 c^4-3 a^6 b^6 c^4-16 a^4 b^8 c^4+45 a^2 b^10 c^4-28 b^12 c^4-29 a^10 c^6+12 a^8 b^2 c^6-3 a^6 b^4 c^6-2 a^4 b^6 c^6-25 a^2 b^8 c^6+56 b^10 c^6+15 a^8 c^8-9 a^6 b^2 c^8-16 a^4 b^4 c^8-25 a^2 b^6 c^8-70 b^8 c^8+3 a^6 c^10+26 a^4 b^2 c^10+45 a^2 b^4 c^10+56 b^6 c^10-9 a^4 c^12-25 a^2 b^2 c^12-28 b^4 c^12+5 a^2 c^14+8 b^2 c^14-c^16,-a^16+5 a^14 b^2-9 a^12 b^4+3 a^10 b^6+15 a^8 b^8-29 a^6 b^10+25 a^4 b^12-11 a^2 b^14+2 b^16+8 a^14 c^2-25 a^12 b^2 c^2+26 a^10 b^4 c^2-9 a^8 b^6 c^2+12 a^6 b^8 c^2-35 a^4 b^10 c^2+34 a^2 b^12 c^2-11 b^14 c^2-28 a^12 c^4+45 a^10 b^2 c^4-16 a^8 b^4 c^4-3 a^6 b^6 c^4+12 a^4 b^8 c^4-35 a^2 b^10 c^4+25 b^12 c^4+56 a^10 c^6-25 a^8 b^2 c^6-2 a^6 b^4 c^6-3 a^4 b^6 c^6+12 a^2 b^8 c^6-29 b^10 c^6-70 a^8 c^8-25 a^6 b^2 c^8-16 a^4 b^4 c^8-9 a^2 b^6 c^8+15 b^8 c^8+56 a^6 c^10+45 a^4 b^2 c^10+26 a^2 b^4 c^10+3 b^6 c^10-28 a^4 c^12-25 a^2 b^2 c^12-9 b^4 c^12+8 a^2 c^14+5 b^2 c^14-c^16 : : 

 

= lies on these lines: {2,3}, {195,11671}, {930,24573}, {1263,25044}, {6343,20424}, {10627,20327}, {15345,20414}

= reflection of X(i) in X(j) for these {i,j}: {3,10285}, {4,20120}, {20,14142}, {10205,5501}, {10627,20327}, {15345,20414}, {27868,20030}

Best regards,
Peter Moses.

 

HYACINTHOS 28647

[Anteas P. Hatzipolakis]:

 

Let ABC be a triangle and L the Euler line.

Denote:

(Na), (Nb), (Nc) = the NPCs of OBC, OCA, OAB, resp.

A', B', C' = the midpoints of AO, BO, CO, resp.

The parallel to L through A' intersects again (Nb), (Nc) at Ab, Ac, resp.

The parallel to L through B' intersects again (Nb), (Na) at Bc, Ba, resp.

The parallel to L through C' intersects again (Nc), (Nb) at Ca, Cb, resp.

Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.

The centroid of MaMbMc lies on the L.

[Peter Moses]:

Hi Antreas,

a^10 b^2+2 a^8 b^4-9 a^6 b^6+5 a^4 b^8+4 a^2 b^10-3 b^12+a^10 c^2-8 a^8 b^2 c^2+10 a^6 b^4 c^2+7 a^4 b^6 c^2-17 a^2 b^8 c^2+7 b^10 c^2+2 a^8 c^4+10 a^6 b^2 c^4-24 a^4 b^4 c^4+13 a^2 b^6 c^4-b^8 c^4-9 a^6 c^6+7 a^4 b^2 c^6+13 a^2 b^4 c^6-6 b^6 c^6+5 a^4 c^8-17 a^2 b^2 c^8-b^4 c^8+4 a^2 c^10+7 b^2 c^10-3 c^12 : : 

= lies on these lines: {2,3}, {523,23332}, {2452,23291}, {9530,24930}, {11550,16319}

Best regards,
Peter Moses.