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