HYACINTHOS 28457

[Peter Moses]:

 

Hi Antreas,

[APH]:

> I did not find this in your search files in CTC:
> X(22335) lies on K115 

 

That is because X(22335), (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^6-3 a^4 b^2-3 a^2 b^4+3 b^6-5 a^4 c^2-a^2 b^2 c^2-5 b^4 c^2+a^2 c^4+b^2 c^4+c^6) (3 a^6-5 a^4 b^2+a^2 b^4+b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2-3 a^2 c^4-5 b^2 c^4+3 c^6)::, does not lie on K115!
It does lie on these lines {4,8254},{381,3459},{546,1141},{1263,3574},{1487,3850},{3845,15619} and the conic (A, B, C X(4), X(5)), which means that it's isog. should be on at least one decent line.

= ISOGONAL CONJUGATE OF  X(22335)

a^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^6+a^4 b^2-5 a^2 b^4+3 b^6+a^4 c^2-a^2 b^2 c^2-3 b^4 c^2-5 a^2 c^4-3 b^2 c^4+3 c^6)::
= lies on these lines: {3,54},{140,1141},{252,549},{546,7604},{4994,18559},{12026,19268},{13367,20574}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 54, 1157), (3, 16762, 12307), (10610, 18016, 3).
= isogonal conjugate of X(22335)

 

I see that K115 is a bit bereft of points ... here is one:

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= ANTICOMPLEMENT OF X(15345)  

b^2 c^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4)::
= lies on the conic (A,B,C,X(4),X(5)), the cubics K028, K115, K526 and these lines: {2,3459},{3,252},{4,93},{17,11082},{18,11087},{26,1601},{30,15619},{343,565},{566,2165},{631,18016},{1263,21230},{1487,1656},{1658,21394},{2937,11816},{2962,18395},{2980,17714},{5449,6368},{5965,20414},{10095,17500},{12044,23181},{14072,19268},{14247,14265},{20424,22335}.
= anticomplement X(15345).
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (93, 11140, 3519), (252, 930, 3), (3519, 18370, 4).
= X(i)-cross conjugate of X(j) for these (i,j): {233, 324}, {1209, 5}, {6368, 930}.
= X(i)-isoconjugate of X(j) for these (i,j): {49, 2190}, {54, 2964}, {1994, 2148}, {2167, 2965}, {2169, 3518}.
= cevapoint of X(3519) and X(21975).
= crosspoint of X(11140) and X(20572).
= barycentric product X(i)X(j) for these {i,j}: {{5, 11140}, {93, 343}, {216, 20572}, {311, 2963}, {324, 3519}, {930, 18314}, {2962, 14213}}.
= barycentric quotient X(i)/X(j) for these {i,j}: {{5, 1994}, {51, 2965}, {53, 3518}, {93, 275}, {216, 49}, {233, 1493}, {311, 7769}, {930, 18315}, {1487, 288}, {1953, 2964}, {2962, 2167}, {2963, 54}, {3519, 97}, {10412, 2413}, {11140, 95}, {12077, 1510}, {20572, 276}}.
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which leads to another point on {3,54}:

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 = X(2)X(252)∩X(3)X(54)

a^4 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4)::
= lies on the cubic K009 and these lines: {2,252},{3,54},{4,137},{5,23338},{24,3432},{32,14586},{49,15787},{96,7578},{140,24305},{184,20574},{186,15620},{288,13472},{1147,15958},{2120,3567},{3090,7604},{3484,6241},{3574,8154},{8254,14143},{8884,18559},{10285,24385},{11671,24144},{11815,15454}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54, 1157, 3), (1493, 6150, 15345), (6150, 15345, 3).
= isogonal of the anticomplement X(15345).
= X(i)-Ceva conjugate of X(j) for these (i,j): {288, 14533}, {933, 1510}, {1166, 54}.
= X(i)-isoconjugate of X(j) for these (i,j): {{5, 2962}, {252, 1087}, {564, 12044}, {930, 2618}, {1953, 11140}, {2963, 14213}}.
= crosssum of X(3519) and X(21975).
= barycentric product X(i)X(j) for these {i,j}: {{49, 275}, {54, 1994}, {95, 2965}, {97, 3518}, {288, 1493}, {1510, 18315}, {2167, 2964}, {15721, 16306}}.
= barycentric quotient X(i)/X(j) for these {i,j}: {{49, 343}, {54, 11140}, {275, 20572}, {1510, 18314}, {1994, 311}, {2148, 2962}, {2964, 14213}, {2965, 5}, {3518, 324}, {8882, 93}, {14533, 3519}, {14586, 930}}.
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Best regards,
Peter Moses.