HYACINTHOS 21959

[Antreas P. Hatzipolakis]:

> Let ABC be a triangle, P, P* two isogonala conjugate points.
>
> Denote:
>
> Ra = radical axis of (NPC_PBC), (NPC_P*BC)
>
> Rb = radical axis of (NPC_PCA), (NPC_P*CA)
>
> Rc = radical axis of (NPC_PAB), (NPC_P*AB)
>
> Which is the locus of P such that Ra,Rb,Rc are concurrent?
> The entire plane?
>
> Antreas

[Barry Wolk]:

This was already answered, and is the whole plane.
Another way of stating this result is:
 
The Poncelet points of the quads BCPP*, CAPP*, ABPP*
form a triangle which is perspective to the medial triangle.
--
Barry Wolk

HYACINTHOS 21942

 

 [Antreas P. Hatzipolakis]:

> > Let ABC be a triangle, P, P* two isogonal conjugate points.
> >
> > Denote:
> >
> > Ra = radical axis of (NPC_PBC), (NPC_P*BC)
> >
> > Rb = radical axis of (NPC_PCA), (NPC_P*CA)
> >
> > Rc = radical axis of (NPC_PAB), (NPC_P*AB)
> >
> > Which is the locus of P such that Ra,Rb,Rc are concurrent?
> > The entire plane?
>
> I wrote this without making a figure. I had in mind
> the points P,P* = O,H: the triangles HBC,HCA, HAB
> share the same NPC, the NPC of ABC, and the NPCs of OBC, OCA, OAB
> concur at the Poncelet point of O wrt ABC, lying on the NPC of ABC.
> Therefore the radical axes in question are concurrent for
> that points.
> Now I make a figure with P,P* = G,K and it seems that
> the radical axes are again concurrent.
>
> See the figure:
>
> http://anthrakitis.blogspot.gr/2013/04/radical-axes-of-npcs.html
>
> Is it true? And if yes, which is the point of concurrence?
>
> In general???
>
> APH

[Randy Hutson]:

Antreas,

I tried various specific pairs of isogonal points, and finally for an
arbitrarily random pair, and it appears the locus is the entire plane. This
does provide an interesting mapping. Some specifics for the concurrence
point for (P,P*):

(G,K): non-ETC -0.956613251489256, which is also the Hyacinthos #16741/16782
homothetic center for line X(2)X(6), and the centroid of (degenerate) pedal
triangle of X(111).

(O,H): X(125)

(N,X54): non-ETC 4.975239945739461

(X7,X55): non-ETC 3.097837435698617

(PU(1)): non-ETC 4.098938269094193, which is the center of conic
{A,B,C,X(99),PU(37)}, complement of X(1916), and anticomplement of X(2023),
and lies on lines 2,694 3,76.

(foci of Steiner inellipse): non-ETC 0.632166489381459, which is the center
of the circumconic through the isotomic conjugates of the foci of the
Steiner inellipse; also the crosssum of X(6) and X(1380), the crosspoint of
X(2) and X(3414); also the complement of the trilinear pole of major axis of
the Steiner eliipses (line X(2)X(1341)); lies on line X(2)X(6) and on the
Steiner inellipse.

(foci of orthic inconic): non-ETC 4.706821577388139, which is the isotomic
conjugate of the polar conjugate of X(1313).

I think this is certainly worth exploring further.

Best regards,
Randy

[Cesar Lozada]:

The trilinear transformation of P( u : v : w ) is :



( f(u, v, w, a, b, c, A, B, C) : f(v, w, u, b, c, a, B, C, A) : f( w, u, v,
c, a, b, C, A, B) )



where



f(u, v, w, a, b, c, A, B, C) = u*(v^2 - w^2)*[u*( v^2 - w^2) - v*(w^2 -
u^2)*cos(C) - w*( u^2 - v^2)*cos(B) ] /a



whose inverse doesn’t seem to be easily calculable.



Regards

Cesar Lozada

HYACINTHOS 10039

Dear Paul,

> [APH]: Let ABC be a triangle and P a point.
> The line AP intersects the circumcircle of triangle
> PBC at A' [other than P].
> Let A" be the orthogonal projection of A' on BC.
> Similarly define B" and C".
>
> [FvL]: Triangle A"B"C" is always perspective to ABC, and the mapping
> of P to this perspector is called by Antreas the REHHAGEL mapping.
> What if we take the desmic mate A1B1C1 of A'B'C' and the orthogonal
> projections of A1B1C1 to ABC as A*B*C*? Is A*B*C* perspective to ABC?
>
> [PY]: If P = (u:v:w), A*B*C* is perspective with ABC at
>
> Q = (1/(-a^2S_A vw + b^2S_B wu + c^2S_c uv + b^2c^2 u^2) : ... : ...).
>
> Here are some examples:
>
> P Q
> -----------------------------
> I, X(36) X(8)
> X(4), X(186) X(68)
>
> *** Note that X(36) is the inverse of I in the circumcircle, and X
> (186) is that of X(4). Also, for P = X(15), X(16) [isodynamic
> points], Q = G, the centroid.
>
>
> This point Q(P) has appeared before. Nik [Hyacinthos 6325] has found
> that the reflection of the pedal triangle of P in its own circumcenter
> is perspective with ABC at Q(P). It is indeed true that inverses in
> the circumcircle have the same Q.

Thank you very much!

Let us call A'B'C' the CircleCevian triangle of P. Because A'B'C' is a
Jacobi-triangle as Darij pointed out, A1B1C1 is just the CircleCevian
triangle of P*.
This means that the REHHAGEL mapping is isogonal conjugacy followed by Nik's
mapping (which I was prepared to call MICHELS mapping - MICHELS was coach of
Dutch soccer teams winning the Euro Champs in 1988 and being runner up of
the World Champs of 1974).

Note that A'A1 // B'B1 // C'C1 // PP*

Kind regards,
Sincerely,
Floor.

HYACINTHOS 29647

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, A'B'C' the pedal triangle of I and (Ia), (Ib), (Ic) the excircles.

Denote:

(Na), (Nb), (Nc) = the NPCs of IBC, ICA, IAB, resp.

Ra = the radical axis of (Na), (Ia)
Rb = the radical axis of (Nb), (Ib)
Rc = the radical axisof (Nc), (Ic)

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

1. A'B'C', A*B*C*
2. IaIbIc, A*B*C*
3. NaNbNc, A*B*C*
are orthologic.

Orthologic centers?


[César Lozada]:

 

 

1)

A*B*C* à A’B’C’ = MIDPOINT OF X(9782) AND X(32635)

= (b^2+6*b*c+c^2)*a^2+12*(b+c)*b*c*a-(b^2-c^2)^2 : : (barys)

= 8*X(5)-3*X(13865), 5*X(1698)-X(5506), X(5557)+9*X(19875), 7*X(9780)+X(9782), 7*X(9780)-X(32635), 3*X(10172)-X(34198)

= lies on these lines: {2, 3303}, {5, 40}, {7, 12}, {10, 354}, {11, 3634}, {20, 26040}, {55, 17552}, {142, 3983}, {442, 3828}, {443, 9657}, {474, 31157}, {495, 11034}, {496, 19872}, {528, 17536}, {548, 5251}, {631, 4413}, {993, 17583}, {1210, 12620}, {1329, 18231}, {1574, 31462}, {2550, 9670}, {2551, 9656}, {2886, 19877}, {3058, 16842}, {3214, 17245}, {3526, 19854}, {3614, 3841}, {3649, 3740}, {3698, 11362}, {4208, 31141}, {4301, 25917}, {4309, 11108}, {4317, 9708}, {4421, 31259}, {4866, 5557}, {4999, 9342}, {5067, 31245}, {5084, 9671}, {5259, 6154}, {5260, 15326}, {5298, 17531}, {5657, 7958}, {5787, 10857}, {6057, 28612}, {6067, 7080}, {6174, 6675}, {7173, 33108}, {7486, 31246}, {8582, 10177}, {8715, 17590}, {9709, 31452}, {9844, 10395}, {9940, 12619}, {9956, 22798}, {10172, 34198}, {12616, 12671}, {12623, 12866}, {16239, 31235}, {16408, 31494}, {17527, 19876}, {17559, 31140}

= midpoint of X(9782) and X(32635)

= X(1173)-of-4th Euler triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 17529, 15888), (3826, 9711, 4197), (3826, 9780, 12), (4197, 9711, 12), (4197, 9780, 9711), (4413, 19855, 24953), (5084, 31420, 9671), (8728, 19875, 21031)

= [ 2.7503993823312540, 1.7039833043620100, 1.1915686331962550 ]

 

A’B’C’ à A*B*C* = X(1)X(550) ∩ X(7)X(12)

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

= X(1)-3*X(5557), 7*X(9780)-15*X(9782), 7*X(9780)-5*X(32635), 3*X(9782)-X(32635)

= lies on these lines: {1, 550}, {7, 12}, {11, 11544}, {57, 5506}, {65, 3625}, {145, 5434}, {553, 1125}, {1071, 31673}, {1317, 4298}, {3337, 3652}, {3634, 3982}, {3679, 5586}, {4031, 19862}, {4355, 10944}, {4860, 7965}, {5433, 21454}, {5708, 7173}, {6797, 11570}, {9776, 28647}, {11495, 30340}, {11551, 13624}, {11684, 26842}, {12690, 33667}, {14100, 15009}, {16006, 18483}, {17718, 31425}, {23958, 31260}

= X(1173)-of-intouch triangle

= X(2889)-of-inverse-in-incircle triangle

= [ 0.6885865080000078, 0.7685580933936743, 2.7907766443272150 ]

 

2)

A*B*C* à IaIbIc = A*B*C* à A’B’C’

IaIbIc à A*B*C* = X(5506)

 

3)

A*B*C* à NaNbNc = X(11)X(3634) ∩ X(119)X(12811)

= (b^2-10*b*c+c^2)*a^5-(b+c)^3*a^4-(2*b^4+2*c^4+(7*b^2-58*b*c+7*c^2)*b*c)*a^3+2*(b+c)*(b^4-8*b^2*c^2+c^4)*a^2+(b^2+17*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : : (barys)

= lies on these lines: {11, 3634}, {119, 12811}, {1156, 5852}, {1484, 5535}

= [ 2.5358708601591540, 5.0415177863158530, -1.0200189979231290 ]

 

NaNbNc à A*B*C* = X(5506)

 

 

César Lozada

HYACINTHOS 29640

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denite:

Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.

A' = BC /\ NbNc
B' = CA /\ NcNa
C' = AB /\ NaNb

A', B',C' are collinear.

Which line is A'B'C' ? (trilinear polar of which point?)


[Peter Moses]:


Hi Antreas,

5*a^12 - 20*a^10*b^2 + 31*a^8*b^4 - 24*a^6*b^6 + 11*a^4*b^8 - 4*a^2*b^10 + b^12 - 20*a^10*c^2 + 32*a^8*b^2*c^2 - 4*a^6*b^4*c^2 - 16*a^4*b^6*c^2 + 16*a^2*b^8*c^2 - 8*b^10*c^2 + 31*a^8*c^4 - 4*a^6*b^2*c^4 + 7*a^4*b^4*c^4 - 12*a^2*b^6*c^4 + 23*b^8*c^4 - 24*a^6*c^6 - 16*a^4*b^2*c^6 - 12*a^2*b^4*c^6 - 32*b^6*c^6 + 11*a^4*c^8 + 16*a^2*b^2*c^8 + 23*b^4*c^8 - 4*a^2*c^10 - 8*b^2*c^10 + c^12  :  :

= lies on this line: {2, 6}


Best regards,
Peter Moses.

HYACINTHOS 29637

[Kadir Aktintas]

Let ABC be a triangle.

Denote:

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

G' = the centroid of NaNbNc

Ga, Gb, Gc = the reflections of of G' in NbNc, NcNa, NaNb, resp.

 

ABC, GaGbGc are perspective.

 

 

[Ercole Suppa]

 

The perspector of ABC, GaGbGc is the point

 

X = X(8)X(31870) ∩ X(9)X(5886)

 

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

 

= lies on the Feuerbach circumhyperbola and these lines: {8,31870}, {9,5886}, {21,13464}, {79,12675}, {90,11522}, {946,1156}, {1537,3065}, {4866,30315}, {5551,10806}, {5665,18990}, {5715,33576}, {5882,17097}, {6596,19907}, {7317,10597}, {7319,26332}, {11496,15446}, {11604,12757}

 

= isogonal conjugate of X*

 

= (6-9-13) search numbers [-0.7839722700258233846, -0.8713748285930834830, 4.6057573340222784612]

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

 

X* = ISOGONAL CONJUGATE OF X = X(1)X(3) ∩ X(21)X(5882)

 

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

 

= lies on these lines: {1,3}, {21,5882}, {100,10165}, {104,33812}, {140,6174}, {355,5259}, {392,6326}, {405,5881}, {411,13464}, {498,6978}, {515,1621}, {519,1006}, {551,6905}, {581,3915}, {631,8715}, {632,10943}, {944,5248}, {952,5251}, {993,7967}, {1001,5587}, {1012,4428}, {1125,6946}, {1283,30285}, {1479,6982}, {1483,5288}, {1953,5011}, {2267,2323}, {2302,22356}, {2772,14094}, {2800,18444}, {2975,13607}, {3058,6907}, {3090,3825}, {3146,10587}, {3149,9624}, {3523,11240}, {3525,10806}, {3529,10532}, {3584,6882}, {3616,6796}, {3624,11499}, {3628,26470}, {3651,4301}, {3655,6914}, {3679,6883}, {3871,6684}, {3884,21740}, {4187,20400}, {4304,12119}, {4309,6850}, {4853,11517}, {4857,6842}, {4863,26446}, {5047,24987}, {5079,18544}, {5231,5687}, {5250,5693}, {5270,7491}, {5284,10175}, {5315,5396}, {5398,16474}, {5657,25439}, {6419,26458}, {6420,26464}, {6827,10056}, {6830,10197}, {6853,24387}, {6875,8666}, {6891,31452}, {6911,25055}, {6916,10385}, {6954,10072}, {6985,11522}, {6986,11362}, {6992,11239}, {7411,28194}, {7412,23710}, {7420,18613}, {7489,28204}, {7580,31162}, {7701,12680}, {7741,26487}, {7988,18491}, {7989,18518}, {8227,11500}, {9956,25542}, {10303,10527}, {10386,11826}, {10541,12595}, {10597,17538}, {11024,17572}, {11230,18524}, {11231,12331}, {12672,16132}, {13218,15020}, {14217,33593}, {14869,32214}, {15172,15908}, {15254,18908}, {15325,21155}, {15888,31789}, {16842,17619}, {17531,24541}, {17536,31399}, {17857,31435}, {19546,29640}, {21628,21669}

 

= isogonal conjugate of X

 

= reflection of X(7688) in X(15931)

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,10267,10902}, {1,10268,12704}, {1,10902,11012}, {1,16208,5709}, {3,3303,7982}, {3,15178,5563}, {55,3576,2077}, {3303,11510,33925}, {3303,33925,1}, {3428,6767,16200}, {3576,12703,30503}, {5010,30392,10269}, {10246,32613,36}, {10267,16202,1}, {10267,24299,14798}

 

= (6-9-13) search numbers [3.9823977416477836882, 3.5829465061622586991, -0.6778666723500078523]

 

 

Best regards,

Ercole Suppa

HYACINTHOS 29636

[Kadir Altintas]

Let ABC be a triangle with circumcenter O.

Denote:
Ga, Gb, Gc = the centroids of OBC, OCA, OAB, resp.. 
N' = the NPC center of GaGbGc.
Na, Nb, Nc = the reflections of N' in GbGc, GcGa, GaGb, resp.

Prove: ANa, BNb, CNc concur at a point X.


[Ercole Suppa]


X = X(4)X(2889) ∩ X(5)X(14483) =

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

= (5 S^2+SC^2) (SB+SC-SW) (-4 S^2+SB SC-SB SW+SC^2-SC SW) : : (barys)

= lies on the Jerabek circumhyperbola and these lines: {4,2889}, {5,14483}, {6,3411}, {20,11738}, {49,1176}, {54,549}, {64,3534}, {74,548}, {185,13623}, {265,1216}, {382,14490}, {1173,3628}, {3426,17800}, {3431,15717}, {3519,3917}, {3521,5562}, {3527,5055}, {3856,14487}, {4846,18436}, {5072,11850}, {7486,14491}, {10303,13472}, {10304,11270}, {10627,15108}, {11559,12121}, {13754,14861}, {15644,16620}, {15704,16659}, {15749,18531}, {15750,18532}

= isogonal conjugate of X*

= (6-9-13) search numbers [5.5986005659038030040, 3.2559808617312889554, -1.1974456066313454063]

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

X* = isogonal conjugate of X = EULER LINE INTERCEPT OF X(6)X(15580) =

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

= SB SC (SB+SC) (-4 S^2-SB SC+SB SW+SC SW-SW^2) : : (barys)

 As a point on the Euler X* has Shinagawa coefficients {-4 f, 5 e + 4 f}

= lies on these lines: {2,3}, {6,15580}, {32,33885}, {51,1199}, {54,1495}, {64,11738}, {74,13474}, {93,32085}, {107,13597}, {110,5446}, {143,10540}, {155,15110}, {156,1994}, {184,9781}, {185,12112}, {232,5041}, {323,10263}, {389,14157}, {569,26881}, {578,26882}, {1056,10046}, {1058,10037}, {1112,2914}, {1173,13366}, {1179,6344}, {1204,11455}, {1216,15107}, {1629,11816}, {1829,33179}, {1831,6198}, {1843,5097}, {1968,10986}, {3060,10539}, {3085,9673}, {3086,9658}, {3199,5008}, {3431,17821}, {3527,26864}, {3563,7953}, {3567,6759}, {3817,9626}, {5102,7716}, {5603,8185}, {5890,26883}, {6242,22750}, {6403,11470}, {7592,17810}, {7687,32340}, {7689,11439}, {7713,16200}, {7967,11365}, {8718,9729}, {8884,11815}, {9590,18483}, {9591,10175}, {9609,31404}, {9625,19925}, {9700,31415}, {9707,10982}, {9713,31418}, {9777,14530}, {9798,10595}, {10095,11817}, {10117,15081}, {10282,15033}, {10575,15053}, {10596,26309}, {10597,26308}, {10984,15024}, {11002,12161}, {11270,14490}, {11278,31948}, {11423,15004}, {11424,11464}, {11438,12290}, {11440,16194}, {11451,13336}, {11456,31860}, {11491,20988}, {11550,26917}, {11572,14644}, {12022,15873}, {12241,12254}, {12310,20125}, {12325,31831}, {13339,32205}, {13353,13364}, {13419,25739}, {13451,14627}, {13472,17809}, {13567,16659}, {13568,32111}, {14683,32358}, {14853,20987},{15052,18436}, {15062,32110}, {15513,33880}, {15749,18532}, {16534,25714}, {16655,26879}, {18912,31383}

= isogonal conjugate of X

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,7517,12088}, {3,4,13596}, {4,24,3520}, {4,25,3518}, {4,186,14865}, {4,3517,17506}, {4,3518,186}, {4,3542,7577}, {4,6143,15559}, {4,7487,18559}, {4,13619,1885}, {4,14940,427}, {4,21844,1593}, {5,23,7512}, {5,7512,7550}, {5,18378,23}, {20,14002,7506}, {22,7529,3090}, {24,378,15750}, {24,1593,21844}, {24,1598,4}, {24,3520,186}, {24,10594,1598}, {25,1598,24}, {25,5198,3517}, {25,10301,23}, {25,10594,4}, {51,1614,1199}, {186,26863,4}, {235,7576,4}, {235,7715,7576}, {378,5198,4}, {378,15750,23040}, {378,23040,3520}, {382,12106,22467}, {382,22467,7464}, {403,6756,4}, {428,1594,4}, {428,21841,1594}, {468,15559,6143}, {546,2070,14118}, {1495,10110,54}, {1593,21844,3520}, {1596,6240,4}, {1656,17714,6636}, {1658,3843,7527}, {1906,18560,4}, {1995,7387,631}, {3199,10312,8744}, {3199,10985,10312}, {3517,5198,378}, {3517,15750,24}, {3518,3520,24}, {3518,10594,26863}, {3518,26863,14865}, {3520,17506,23040}, {3542,6995,4}, {3567,6759,15032}, {3628,13564,15246}, {3855,7556,7503}, {3861,7575,14130}, {5020,10323,3525}, {5899,18369,140}, {7486,7492,7516}, {7503,9714,7556}, {7506,7530,20}, {7517,13861,2}, {7545,18378,5}, {10263,18350,323}, {11799,31830,34007}, {13564,21308,3628}, {15750,23040,17506}, {17928,18534,3529}

= (6-9-13) search numbers [-1.2041210079187199799, -2.0704644291937978833, 5.6298110903887253207]


Best regards,
Ercole Suppa

HYACINTHOS 29631

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the medial triangle.

Denote:

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

Oa, Ob, Oc = the circumcenters of OBC, OCA, OAB, resp.
Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.

Goa, Gob, Goc = the centroids of A'A"Oa, B'B"Ob, C'C"Oc, resp.  
Gna, Gnb, Gnc = the centroids of A'A"Na, B'B"Nb, C'C"Nc, resp.

1. The circumcenter of GoaGobGoc lies on the Euler line
2. The NPC center of GnaGnbGnc lies on the Euler line.

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

Let ABC be a triangle and A'B'C' the medial triangle.

Denote:

A", B", C"  the midpoints of AN, BN, CN, resp.

Oa, Ob, Oc = the circumcenters of NBC, NCA, NAB, resp.

Ga, Gb, Gc = the centroids of OaA'A", ObB'B", OcC'C", resp.

3. The NPC center of GaGbGc lies on the Euler line


[Peter Moses]:


Hi Antreas,

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

1).
= 4*a^10 - 9*a^8*b^2 + 2*a^6*b^4 + 8*a^4*b^6 - 6*a^2*b^8 + b^10 - 9*a^8*c^2 + 12*a^6*b^2*c^2 - 6*a^4*b^4*c^2 + 6*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 6*a^4*b^2*c^4 + 2*b^6*c^4 + 8*a^4*c^6 + 6*a^2*b^2*c^6 + 2*b^4*c^6 - 6*a^2*c^8 - 3*b^2*c^8 + c^10 : :

= lies on these lines: {2, 3}, {343, 1511}, {2883, 32210}, {3917, 16223}, {5642, 11562}, {5663, 10192}, {6697, 11645}, {6699, 20773}, {10182, 13754}, {10193, 14915}, {10282, 20191}, {12359, 32171}, {16226, 32352}, {16252, 32138}, {17821, 32140}

= midpoint of X(i) and X(j) for these {i,j}: {2,18324}, {3,10201}, {549,34351}, {14070,18281}, {15331,34330}
= reflection of X(i) in X(j) for these {i,j}: {5,34330}, {10201,10020}, {15761,10201}, {18566,5066}, {34330,10125}, {34351,15330}

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

2).
= 6*a^10 - 15*a^8*b^2 + 6*a^6*b^4 + 12*a^4*b^6 - 12*a^2*b^8 + 3*b^10 - 15*a^8*c^2 + 26*a^6*b^2*c^2 - 13*a^4*b^4*c^2 + 11*a^2*b^6*c^2 - 9*b^8*c^2 + 6*a^6*c^4 - 13*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 6*b^6*c^4 + 12*a^4*c^6 + 11*a^2*b^2*c^6 + 6*b^4*c^6 - 12*a^2*c^8 - 9*b^2*c^8 + 3*c^10 : :

= lies on these lines: {2, 3}, {3410, 22251}

= midpoint of X(i) and X(j) for these {i,j}: {549,34331}, {15330,18281}, {15332,18568}
= reflection of X(25401) in X(34331) 

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

3).
= 2*a^16 - 17*a^14*b^2 + 65*a^12*b^4 - 139*a^10*b^6 + 175*a^8*b^8 - 127*a^6*b^10 + 47*a^4*b^12 - 5*a^2*b^14 - b^16 - 17*a^14*c^2 + 90*a^12*b^2*c^2 - 171*a^10*b^4*c^2 + 98*a^8*b^6*c^2 + 89*a^6*b^8*c^2 - 138*a^4*b^10*c^2 + 51*a^2*b^12*c^2 - 2*b^14*c^2 + 65*a^12*c^4 - 171*a^10*b^2*c^4 + 96*a^8*b^4*c^4 + 29*a^6*b^6*c^4 + 72*a^4*b^8*c^4 - 123*a^2*b^10*c^4 + 32*b^12*c^4 - 139*a^10*c^6 + 98*a^8*b^2*c^6 + 29*a^6*b^4*c^6 + 38*a^4*b^6*c^6 + 77*a^2*b^8*c^6 - 94*b^10*c^6 + 175*a^8*c^8 + 89*a^6*b^2*c^8 + 72*a^4*b^4*c^8 + 77*a^2*b^6*c^8 + 130*b^8*c^8 - 127*a^6*c^10 - 138*a^4*b^2*c^10 - 123*a^2*b^4*c^10 - 94*b^6*c^10 + 47*a^4*c^12 + 51*a^2*b^2*c^12 + 32*b^4*c^12 - 5*a^2*c^14 - 2*b^2*c^14 - c^16 : :

= lies on this line: {2, 3}.

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

Best regards,
Peter Moses.

 

HYACINTHOS 29626

[Kadir Altintas]:

 

 

Let ABC be a triangle and P a point.

 

Denote:

 

Ga, Gb, Gc = the centroids of PBC, PCA, PAB, resp. 

K' = the symmedian point of GaGbGc. 

Ka, Kb, Kc = the reflections of K' in GbGc, GcGa, GaGb, resp.

 

Which is the locus of P such that ABC and KaKbKc are perspective ?

 

 

[Ercole Suppa]:

 

 

The locus of P such that ABC and KaKbKc are perspective is {Linf}∪{q2 = conic through X(6),X(3413),X(3414),X(11477),X(15534),X(32935)}

 

If P = (x:y:z) the perspector is the point Q = Q(P) = (2 b^2 x-c^2 x+2 b^2 y-c^2 y+a^2 z+3 b^2 z) (-b^2 x-3 c^2 x+a^2 y-2 c^2 y+a^2 z-2 c^2 z)  : :   (barys)

 

*** Pairs {P = X(i)∈ q1, Q(P) = X(j)} for these {i,j}: {6,76},{3413,3413},{3414,3414},{11477,262},{15534,2}

 

*** Some points:

 

Q1= Q(X(32935)) = MIDPOINT OF X(7985) AND X(9902) =

 

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

 

= X[7985]+X[9902]

 

= lies on the Kiepert circumhyperbola and these lines: {2,726}, {10,22036}, {76,4066}, {226,4135}, {516,14458}, {519,598}, {2321,11599}, {3906,4049}, {3993,21101}, {3994,30588}, {4134,14839}, {4444,30519}, {4709,13576}, {6625,17760}, {7985,9902}, {11167,17132}}

 

= isogonal conjugate of Q2

= midpoint of X(7985) and X(9902)

 

= ETC search numbers: [5.5609180703422668398, 1.0923585942098356609, 0.3178386534503697195]

 

 

Q2 = Q1* (isogonal conjugate of Q1) = X(3)X(6) ∩ X(81)X(10789) =

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

= lies on these lines: {3,6}, {81,10789}, {106,11636}, {110,727}, {385,24267}, {560,595}, {741,30554}, {1357,1412}, {2712,32694}, {4653,11364}, {4658,12194}, {6233,17222}, {7787,25526}, {21793,23095}

= isogonal conjugate of Q1

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {58,33628,1326}, {1333,5009,58}

= ETC search numbers: [0.1694755862341742547, 0.8627568409924841824, 2.9651517829584250131]

 

Best regards 

Ercole Suppa

 

HYACINTHOS 29622

[Kadir Atintas]:

 

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

Denote:

Oa, Ob, Oc = the circumcenters of PBC, PCA, PAB, resp.
Ga, Gb, Gc = the  centroids of PObOc, POcOa, POaOb, resp.

Which is the locus of P such that A'B'C' and GaGbGc are perspective?  

Some perspectors?

 

[César Lozada]:

 

Locus = {Linf} ∪ {circumcircle} ∪ {q6=excentral-circum-degree-6 through ETCs 1, 3, 4}

 

q6: ∑ [ y*z*(-2*(b^2-c^2)*b^2*c^2*x^4+(-c^2*(3*a^4-2*(b^2+3*c^2)*a^2+b^4-2*b^2*c^2+3*c^4)*y+b^2*(3*a^4-2*(3*b^2+c^2)*a^2+3*b^4-2*b^2*c^2+c^4)*z)*x^3+2*a^4*y^3*z*c^2+2*(b^2-c^2)*a^4*y^2*z^2-2*a^4*y*z^3*b^2)] = 0

 

ETC-pairs (P,Q(P)=perspector): {4, 4}, {74, 15055}, {99, 21166}, {107, 23239}, {110, 15035}

 

If P lies on the circumcircle then OQ=(1/3)*OP, ie, Q lies on the circle (O, R/3).

 

Q( X(1) ) = MIDPOINT OF X(1) AND X(3612)

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

= 2*X(1)+X(5217)

= lies on these lines: {1, 3}, {2, 10950}, {4, 15950}, {6, 17440}, {8, 4999}, {11, 2476}, {12, 944}, {21, 2320}, {33, 4214}, {37, 2261}, {45, 22356}, {78, 3711}, {80, 1656}, {140, 10573}, {145, 5218}, {214, 474}, {226, 9657}, {244, 8572}, {381, 5443}, {382, 18393}, {388, 6840}, {390, 25557}, {392, 1858}, {405, 30144}, {442, 26475}, {497, 2475}, {498, 952}, {515, 10895}, {551, 950}, {632, 11545}, {946, 12953}, {956, 22836}, {958, 3715}, {962, 15338}, {993, 5730}, {1001, 10394}, {1056, 6903}, {1058, 6951}, {1125, 1837}, {1201, 14547}, {1317, 12247}, {1329, 10955}, {1387, 13274}, {1389, 6942}, {1437, 4653}, {1468, 2361}, {1479, 5901}, {1483, 12647}, {1486, 18614}, {1831, 17523}, {1836, 4297}, {1854, 10535}, {1864, 5436}, {2170, 4258}, {2256, 17438}, {2268, 4287}, {2269, 5036}, {2293, 19945}, {2330, 3242}, {2886, 10959}, {2975, 12635}, {3035, 5554}, {3058, 4313}, {3085, 6952}, {3086, 6853}, {3158, 3893}, {3207, 17451}, {3241, 4995}, {3243, 15837}, {3474, 4323}, {3475, 4308}, {3476, 5703}, {3485, 5731}, {3487, 5434}, {3488, 6937}, {3526, 5444}, {3560, 6265}, {3583, 18493}, {3586, 9624}, {3623, 5281}, {3624, 5727}, {3636, 12053}, {3649, 4293}, {3655, 11237}, {3683, 15829}, {3689, 4853}, {3698, 5438}, {3754, 16371}, {3816, 10958}, {3868, 11194}, {3870, 11260}, {3871, 10912}, {3878, 16370}, {3890, 4428}, {3895, 33895}, {3913, 4861}, {3927, 4867}, {3940, 5258}, {4294, 10595}, {4295, 15326}, {4302, 22791}, {4304, 12701}, {4305, 5603}, {4311, 10404}, {4317, 6147}, {4325, 18541}, {4413, 19860}, {4421, 14923}, {4423, 19861}, {4640, 11682}, {4855, 5836}, {4863, 12437}, {4870, 9612}, {5054, 5445}, {5252, 5882}, {5283, 11998}, {5326, 9780}, {5426, 17637}, {5433, 18391}, {5441, 9668}, {5499, 15174}, {5592, 23761}, {5691, 17605}, {5736, 17221}, {5794, 24541}, {5818, 20400}, {5886, 10572}, {6049, 10578}, {6738, 17728}, {6827, 18962}, {6882, 10954}, {7052, 22236}, {7082, 31435}, {7221, 11997}, {7770, 30140}, {7866, 30120}, {7951, 18525}, {7968, 19038}, {7969, 19037}, {7983, 15452}, {9581, 25055}, {9670, 30384}, {9673, 11365}, {9844, 10393}, {10058, 19907}, {10072, 12433}, {10106, 17718}, {10165, 24914}, {10200, 34123}, {10283, 15171}, {10592, 28224}, {10826, 11230}, {10827, 28204}, {11285, 30136}, {11502, 25524}, {11715, 12739}, {11723, 12374}, {11724, 12185}, {11725, 13183}, {11729, 12764}, {11735, 12904}, {12047, 12943}, {12114, 21740}, {12743, 16173}, {13463, 20075}, {13607, 31397}, {13901, 19066}, {13902, 19030}, {13958, 19065}, {13959, 19029}, {15015, 17636}, {15228, 15696}, {17044, 26101}, {17662, 31480}, {18526, 31479}, {21031, 27383}, {21677, 30478}, {22238, 33655}, {23846, 28348}, {24558, 26105}, {28922, 30847}, {28924, 30826}, {30124, 32954}, {31165, 31424}

= midpoint of X(1) and X(3612)

= reflection of X(i) in X(j) for these (i,j): (5217, 3612), (10895, 11375)

= X(3612)-of-anti-Aquila triangle

= X(5217)-of-Mandart-incircle triangle

= X(7547)-of-2nd circumperp triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 3304, 10966), (65, 3576, 5204), (999, 7742, 56), (3057, 17609, 18839), (3338, 5126, 56), (3340, 7987, 1155), (3576, 16193, 56), (5010, 11009, 12702), (8071, 16203, 56), (10267, 22766, 5172), (16193, 31786, 65)

= [ 2.3795882908092240, 2.2594648434366430, 0.9781480714624456 ]

 

Q( X(3) ) = X(5)X(11202) ∩ X(6)X(3515)

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

= (SB+SC)*(16*R^2+SA-5*SW)*(S^2+(16*R^2+SA-6*SW)*SA) : : (barys)

= lies on these lines: {5, 11202}, {6, 3515}, {1147, 18324}, {1493, 23358}, {3292, 22333}, {4550, 10282}, {11821, 15035}, {17821, 33537}

= [ 3.5496134467512130, 2.0629597352178810, 0.5741784590252865 ]

 

Q( X(98) ) = X(3)X(76) ∩ X(20)X(115)

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

= 2*X(3)+X(98), 4*X(3)-X(99), X(3)+2*X(12042), 5*X(3)+X(12188), 7*X(3)-X(13188), 10*X(3)-X(23235), 5*X(3)-2*X(33813), X(4)-4*X(6036), 2*X(4)-5*X(14061), 2*X(98)+X(99), X(98)-4*X(12042), 5*X(98)-2*X(12188), 7*X(98)+2*X(13188), 5*X(98)+X(23235), 5*X(98)+4*X(33813), X(99)+8*X(12042), 5*X(99)+4*X(12188), 7*X(99)-4*X(13188), 5*X(99)-2*X(23235), 8*X(6036)-5*X(14061), 5*X(14061)-4*X(23514)

= lies on these lines: {2, 2794}, {3, 76}, {4, 6036}, {5, 10722}, {20, 115}, {30, 9166}, {35, 10069}, {36, 10053}, {40, 7983}, {74, 15342}, {83, 13335}, {114, 631}, {140, 6033}, {147, 620}, {148, 3522}, {182, 10753}, {186, 30716}, {187, 5999}, {262, 12150}, {315, 8781}, {371, 19055}, {372, 19056}, {376, 671}, {381, 34127}, {385, 18860}, {511, 21445}, {542, 3524}, {543, 10304}, {549, 6054}, {550, 6321}, {648, 14060}, {690, 15055}, {962, 11725}, {1003, 9756}, {1092, 3044}, {1151, 19109}, {1152, 19108}, {1350, 10754}, {1352, 7835}, {1385, 7970}, {1569, 15515}, {1587, 8980}, {1588, 13967}, {1656, 22505}, {1657, 22515}, {1916, 5188}, {2023, 3053}, {2077, 13189}, {2407, 13479}, {2482, 11177}, {2784, 10164}, {2790, 20792}, {3023, 5204}, {3027, 5217}, {3091, 6722}, {3515, 12131}, {3516, 5186}, {3525, 6721}, {3528, 13172}, {3529, 20398}, {3543, 5461}, {3564, 7799}, {3785, 32458}, {3788, 9863}, {3839, 14971}, {3843, 15092}, {3972, 13860}, {4027, 33004}, {4188, 5985}, {4297, 13178}, {5010, 10086}, {5013, 12829}, {5054, 23234}, {5055, 26614}, {5085, 5182}, {5092, 12177}, {5171, 12176}, {5432, 12184}, {5433, 12185}, {5569, 9877}, {5584, 22514}, {5969, 31884}, {5984, 14981}, {5986, 15246}, {6034, 29181}, {6194, 9888}, {6308, 8295}, {6684, 9864}, {6699, 11005}, {6713, 10768}, {7280, 10089}, {7603, 10486}, {7709, 9734}, {7710, 33216}, {7757, 9755}, {7760, 9737}, {7793, 30270}, {7824, 10333}, {7894, 10983}, {7911, 32152}, {7987, 9860}, {8703, 11632}, {8721, 32964}, {8724, 12100}, {9167, 15708}, {9747, 15078}, {9880, 11001}, {10267, 12190}, {10269, 12189}, {10303, 31274}, {10347, 26316}, {10516, 33220}, {10733, 15359}, {10769, 24466}, {10992, 21735}, {11012, 13190}, {11599, 12512}, {12041, 18332}, {12121, 15535}, {12243, 19708}, {12355, 15695}, {13174, 16192}, {13182, 15326}, {13183, 15338}, {14223, 18556}, {14532, 15655}, {15694, 22566}, {15721, 22247}, {15803, 24472}, {16111, 33511}, {20094, 21734}, {21163, 33273}

= midpoint of X(i) and X(j) for these {i,j}: {98, 21166}, {376, 14651}, {14830, 15561}

= reflection of X(i) in X(j) for these (i,j): (4, 23514), (99, 21166), (381, 34127), (671, 14651), (3839, 14971), (5055, 26614), (5182, 5085), (6054, 15561), (14651, 6055), (15561, 549), (21166, 3), (23234, 5054), (23514, 6036)

= circumperp conjugate of X(23235)

= X(5085)-of-1st anti-Brocard triangle

= X(21166)-of-ABC-X3 reflections triangle

= X(23514)-of-anti-Euler triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 98, 99), (3, 12042, 98), (3, 12188, 33813), (4, 6036, 14061), (20, 115, 10723), (40, 11710, 7983), (98, 23235, 12188), (631, 9862, 114), (12188, 33813, 23235), (23235, 33813, 99)

= [ 7.8969073743790900, 8.3655455830454370, -5.7955935560681970 ]

 

Q( X(100) ) = X(3)X(8) ∩ X(20)X(119)

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

= 2*X(3)+X(100), 4*X(3)-X(104), 5*X(3)+X(12331), 7*X(3)-X(12773), X(3)+2*X(33814), X(4)-4*X(3035), X(4)+2*X(24466), 4*X(5)-X(10724), 2*X(10)+X(12119), 2*X(100)+X(104), 5*X(100)-2*X(12331), 7*X(100)+2*X(12773), X(100)-4*X(33814), 5*X(104)+4*X(12331), 7*X(104)-4*X(12773), X(104)+8*X(33814), X(944)+2*X(1145), 2*X(3035)+X(24466), 2*X(4996)+X(11491), 4*X(5690)-X(12531)

= lies on these lines: {2, 5840}, {3, 8}, {4, 3035}, {5, 10724}, {10, 12119}, {11, 631}, {20, 119}, {21, 11231}, {35, 6940}, {36, 10087}, {40, 214}, {80, 6684}, {140, 10738}, {149, 3523}, {153, 3522}, {165, 2800}, {182, 10755}, {371, 19112}, {372, 19113}, {376, 2829}, {404, 5886}, {516, 1519}, {517, 4881}, {528, 3524}, {548, 11698}, {549, 10707}, {550, 10742}, {620, 10768}, {962, 11729}, {1006, 3586}, {1125, 14217}, {1151, 19082}, {1152, 19081}, {1155, 12739}, {1156, 31658}, {1317, 5204}, {1320, 1385}, {1350, 10759}, {1376, 6950}, {1387, 9785}, {1484, 3530}, {1537, 6361}, {1587, 13922}, {1588, 13991}, {1656, 22938}, {1657, 22799}, {1698, 6246}, {1768, 16192}, {1783, 22055}, {1811, 3417}, {1862, 3515}, {2771, 15055}, {2787, 21166}, {2801, 21165}, {2802, 3576}, {2803, 23239}, {3090, 31235}, {3487, 24465}, {3516, 12138}, {3525, 6667}, {3528, 12248}, {3529, 20400}, {3579, 6265}, {3601, 12736}, {3624, 16174}, {3651, 5660}, {3654, 10031}, {3871, 32612}, {4188, 11248}, {4293, 10956}, {4297, 12751}, {4302, 6963}, {4421, 5854}, {5054, 34126}, {5083, 15803}, {5085, 9024}, {5122, 14151}, {5171, 13194}, {5218, 6955}, {5253, 10283}, {5432, 6951}, {5433, 13274}, {5541, 7987}, {5584, 22775}, {5587, 6906}, {5603, 16371}, {5732, 6594}, {5759, 10427}, {5848, 10519}, {5851, 21168}, {5856, 21151}, {6036, 10769}, {6049, 12735}, {6154, 10299}, {6326, 12520}, {6699, 10778}, {6702, 31423}, {6710, 10772}, {6711, 10777}, {6712, 10770}, {6718, 10771}, {6868, 32554}, {6876, 12332}, {6902, 12764}, {6909, 28160}, {6911, 9779}, {6920, 10172}, {6937, 8068}, {6942, 10310}, {6946, 7988}, {6949, 11826}, {6986, 33862}, {7280, 10074}, {7489, 9342}, {7972, 11362}, {7991, 25485}, {8104, 8127}, {8128, 13267}, {8674, 15035}, {9588, 9897}, {10164, 21161}, {10175, 28461}, {10267, 13279}, {10269, 13278}, {10306, 19537}, {10525, 17566}, {11012, 12776}, {11571, 31806}, {12333, 12868}, {12512, 21635}, {12515, 22935}, {12532, 31837}, {12653, 30389}, {12702, 19907}, {12737, 13624}, {12738, 13243}, {12743, 24914}, {12749, 21578}, {12763, 15326}, {13334, 32454}, {13607, 26726}, {13912, 19078}, {13975, 19077}, {15528, 16209}, {15717, 20095}, {16370, 34122}, {17549, 26446}, {17654, 31787}, {24042, 31263}

= midpoint of X(165) and X(15015)

= reflection of X(i) in X(j) for these (i,j): (5603, 34123), (16173, 10165)

= anticomplement of X(23513)

= X(15035)-of-1st circumperp triangle

= X(15055)-of-2nd circumperp triangle

= X(23515)-of-excentral triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 100, 104), (3, 5690, 5303), (3, 33814, 100), (20, 119, 10728), (40, 214, 10698), (100, 5303, 12531), (631, 13199, 11), (3035, 24466, 4), (5541, 7987, 11715), (22935, 31663, 12515)

= [ 5.3888539744561470, 3.4343794492541270, -1.2241462019405450 ]

 

César Lozada

HYACINTHOS 29618

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, A'B'C' the pedal triangle of O and P a point on the Euler line such that OP/OH = t : number

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

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

Ma, Mb, Mc = the midpoints of A'Na, B'Nb, C'Nc, resp

M1, M2, M3 = the midpoints of A"Ma, B"Mb, C"Mc, resp.

Conjecture:
The P point of M1M2M3 (ie the point P' on the Euler line of M1M2M3 such that O'P'/O'H' = t, where O', H' = the circumcenter, orthocenter of M1M2M3, resp.) lies on the Euler line of ABC.


P = N: Hyacinthos 29611 
P = O: Hyacinthos 29613 

                     

[César Lozada]:

 

Conjecture proved.

 

t’ = OP’/OH =

= 1/8*(3*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))^2*t^6-4*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))^2*t^5+(7*a^12-11*(b^2+c^2)*a^10-(19*b^4-56*b^2*c^2+19*c^4)*a^8+(b^2+c^2)*(46*b^4-91*b^2*c^2+46*c^4)*a^6-(19*b^8+19*c^8+3*b^2*c^2*(15*b^4-41*b^2*c^2+15*c^4))*a^4-(b^4-c^4)*(b^2-c^2)*(11*b^4-45*b^2*c^2+11*c^4)*a^2+(7*b^4+17*b^2*c^2+7*c^4)*(b^2-c^2)^4)*t^4+(a^12-6*(b^2+c^2)*a^10+(15*b^4-b^2*c^2+15*c^4)*a^8-(b^2+c^2)*(20*b^4-27*b^2*c^2+20*c^4)*a^6+(15*b^8+15*c^8+b^2*c^2*(7*b^4-36*b^2*c^2+7*c^4))*a^4-(b^4-c^4)*(b^2-c^2)*(6*b^4+7*b^2*c^2+6*c^4)*a^2+(b^2-c^2)^6)*t^3+(-2*a^12+3*(b^2+c^2)*a^10+2*(b^2-3*c^2)*(3*b^2-c^2)*a^8-(b^2+c^2)*(14*b^4-31*b^2*c^2+14*c^4)*a^6+(6*b^8+6*c^8+b^2*c^2*(17*b^4-45*b^2*c^2+17*c^4))*a^4+(b^4-c^4)*(b^2-c^2)*(3*b^4-17*b^2*c^2+3*c^4)*a^2-(b^2-c^2)^4*(b^2+2*c^2)*(2*b^2+c^2))*t^2+(2*(b^2+c^2)*a^10-(8*b^4-5*b^2*c^2+8*c^4)*a^8+(b^2+c^2)*(12*b^4-19*b^2*c^2+12*c^4)*a^6-(8*b^8+8*c^8+b^2*c^2*(7*b^4-26*b^2*c^2+7*c^4))*a^4+(b^4-c^4)*(b^2-c^2)*(2*b^4+7*b^2*c^2+2*c^4)*a^2+2*(b^2-c^2)^4*b^2*c^2)*t-a^2*b^2*c^2*(2*a^6-2*(b^2+c^2)*a^4-(2*b^4-3*b^2*c^2+2*c^4)*a^2+2*(b^4-c^4)*(b^2-c^2)))/((2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*t-a^2*(a^2-b^2-c^2))/((a^4+(b^2-2*c^2)*a^2-(b^2-c^2)*(2*b^2+c^2))*t-b^2*(a^2-b^2+c^2))/((a^4-(2*b^2-c^2)*a^2+(b^2-c^2)*(b^2+2*c^2))*t-c^2*(a^2+b^2-c^2))

 

i.e,

P’(t) = 3*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))^2*t^6-4*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))^2*t^5+(7*a^12-11*(b^2+c^2)*a^10-(19*b^4-56*b^2*c^2+19*c^4)*a^8+(b^2+c^2)*(46*b^4-91*b^2*c^2+46*c^4)*a^6-(19*b^8+19*c^8+3*b^2*c^2*(15*b^4-41*b^2*c^2+15*c^4))*a^4-(b^4-c^4)*(b^2-c^2)*(11*b^4-45*b^2*c^2+11*c^4)*a^2+(7*b^4+17*b^2*c^2+7*c^4)*(b^2-c^2)^4)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*t^4-(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(7*a^12-10*(b^2+c^2)*a^10-(31*b^4-73*b^2*c^2+31*c^4)*a^8+(b^2+c^2)*(84*b^4-163*b^2*c^2+84*c^4)*a^6-(71*b^8+71*c^8+b^2*c^2*(15*b^4-164*b^2*c^2+15*c^4))*a^4+(b^4-c^4)*(b^2-c^2)*(22*b^4+47*b^2*c^2+22*c^4)*a^2-(b^2-c^2)^6)*t^3+(4*a^16-(61*b^4-70*b^2*c^2+61*c^4)*a^12+(b^2+c^2)*(163*b^4-280*b^2*c^2+163*c^4)*a^10-(180*b^8+180*c^8+(65*b^4-448*b^2*c^2+65*c^4)*b^2*c^2)*a^8+(b^2+c^2)*(86*b^8+86*c^8+(118*b^4-409*b^2*c^2+118*c^4)*b^2*c^2)*a^6-(b^2-c^2)^2*(5*b^8+5*c^8+7*(16*b^4+29*b^2*c^2+16*c^4)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)^3*(9*b^4+10*b^2*c^2+9*c^4)*a^2+(b^2-c^2)^6*(b^2+2*c^2)*(2*b^2+c^2))*t^2+(-4*(b^2+c^2)*a^14+2*(11*b^4-5*b^2*c^2+11*c^4)*a^12-(b^2+c^2)*(50*b^4-81*b^2*c^2+50*c^4)*a^10+4*(15*b^8+15*c^8+b^2*c^2*(2*b^4-33*b^2*c^2+2*c^4))*a^8-4*(b^2+c^2)*(10*b^8+10*c^8+b^2*c^2*(2*b^4-25*b^2*c^2+2*c^4))*a^6+2*(b^2-c^2)^2*(7*b^8+7*c^8+6*b^2*c^2*(4*b^4+5*b^2*c^2+4*c^4))*a^4-(b^4-c^4)*(b^2-c^2)^3*(2*b^4+b^2*c^2+2*c^4)*a^2-2*(b^2-c^2)^6*b^2*c^2)*t+a^2*b^2*c^2*(4*a^10-10*(b^2+c^2)*a^8+2*(2*b^4+9*b^2*c^2+2*c^4)*a^6+(b^2+c^2)*(8*b^4-17*b^2*c^2+8*c^4)*a^4-(b^2-c^2)^2*(8*b^4+9*b^2*c^2+8*c^4)*a^2+2*(b^4-c^4)*(b^2-c^2)^3) : :

 

ETC pairs (P,P’): (2,3628), (4,3850), (5,34420)

 

P’( X(3) ) = P’(t=0) = MIDPOINT OF X(140) AND X(5498)

= 4*a^10-10*(b^2+c^2)*a^8+2*(2*b^4+9*b^2*c^2+2*c^4)*a^6+(b^2+c^2)*(8*b^4-17*b^2*c^2+8*c^4)*a^4-(b^2-c^2)^2*(8*b^4+9*b^2*c^2+8*c^4)*a^2+2*(b^4-c^4)*(b^2-c^2)^3 : : (barys)

= (49*R^2-12*SW)*S^2-(19*R^2-4*SW)*SB*SC : : (barys)

= 3*X(2)+X(10226), 15*X(2)+X(34350), 3*X(3)+X(18567), X(3)+3*X(34331)

= As a point on the Euler line, this center has Shinagawa coefficients (E-48*F, -3*E+16*F)

= lies on these lines: {2, 3}, {15311, 32415}

= midpoint of X(i) and X(j) for these {i,j}: {140, 5498}, {461, 11343}, {10125, 23336}, {18420, 25647}

= reflection of X(i) in X(j) for these (i,j): (3536, 33001), (3628, 12043)

= complement of the complement of X(10226)

= [ 4.2238135009577640, 3.3431510513947140, -0.6232770925770827 ]

 

P’( X(20) ) = P’(t=-1) = MIDPOINT OF X(3) AND X(15948)

= 9*S^4+(16*R^2*(80*R^2-31*SW)-11*SB*SC+44*SW^2)*S^2-4*(4*R^2-SW)*(112*R^2-17*SW)*SB*SC : : (barys)

= lies on these lines: {2, 3}

= midpoint of X(3) and X(15948)

= reflection of X(25450) in X(10691)

= [ 9.1279702749931410, 8.2343705360318430, -6.2729629391883490 ]

 

P’( X(550) ) = P’(t=-1/2) = MIDPOINT OF X(3530) AND X(15949)

= 540*S^4+3*(25*R^2*(805*R^2-284*SW)-156*SB*SC+520*SW^2)*S^2-5*(5*R^2*(2125*R^2-764*SW)+296*SW^2)*SB*SC : : (barys)

= lies on these lines: {2, 3}

= midpoint of X(3530) and X(15949)

= reflection of X(26028) in X(21518)

= [ 6.0041159705419130, 5.1187570381120540, -2.6742208385740800 ]

 

César Lozada