HYACINTHOS 28525

 

[Etienne Rousée]:

 

Clark Kimberling in "Triangle Centers and Central Triangles" has a chapter on Circles (ch. 8). In pp. 226-234 writes about Central Circles, where are listed central circles passing through Parry point X(111), passing through X(186), circles passing more than four indexed centers, central circles passing through two indexed points on the circumcircle and also through the Feuerbach point X(11) and mentions also some named circles (Lester circle etc). 

(Note that the book contains only 400 triangle centers and now are listed in ETC 25566 centers)
There is also a listing with a few centers of these circles. The centers of the circles passing through Parry point have been computed, but how about the centers of the other circles (through X(186) erc)?

For interested readers, who do not have the book, I scanned some pages and uploaded them in my blog:
https://anthrakitis.blogspot.com/2018/09/blog-post.html

 

Recently Rescassol [nickname of Etienne Rousée] posted to the French mathematics forum les-mathematiques.net a possibly new circle:

 

Montrer alors que ces 4 points X(6) ,X(11) ,X(37) ,X(55) sont cocycliques. 
Le centre de leur cercle circonscrit n'est pas dans l'ETC. 
http://www.les-mathematiques.net/phorum/read.php?8,1725446

 

To my knowledge it is a new circle. Which is its center?

 

APH

 

[Peter Moses]:

Hi Antreas,

Excluding the circumcircle, some circles on 6 or more points through X(111):

{{2,15,16,23,110,111,352,353,5638,5639,6141,6142,7598,7599,7601,7602,7711,9138,9147,9153,9156,9157,9158,9162,9163,9212,9213,9978,9980,9998,9999,11199,11673,13114,13242,14660,14704,14705},X(351)}
{{6,111,112,115,187,1560,2079,3569,5000,5001,5523,5913,6032,8105,8106,8426,8427,8428,8429,8430,21397},X(2492)}
{{2,3,6,111,691,5653,9173,9174,9178,11579,11632,11637,11638,14174,14180,14699,14700,15546,15744,21732,21733},X(9175)}
{{2,13,14,111,476,5466,5640,6032,6792,7698,9140,9159,11628,11639,11640,13636,13722,14846,14932},X(8371)}
{{3,23,99,111,2079,2930,5104,11641,11643,14669,14678,14682,15564,18773,18774},X(11616)}
{{3,110,111,187,351,2482,6055,6091,7426,7600,9828,9829,19901,19902},X(9126)}
{{110,111,114,115,399,10276,11258,11637,11639,11641,11642,15367,19164},X(11615)}
{{2,111,182,187,6593,7575,9208,11636,11642,11643,14649,14650,15925},X(11621)}
{{6,111,353,2395,2549,2715,5941,6792,10097,10766,11646},X(0)}
{{6,23,111,381,671,2080,9970,11258,11636,11640,19906},X(11622)}
{{5,23,111,115,827,6593,11638,14885,19140,22105},X(11620)}
{{4,107,111,671,5523,7426,9979,20410,24007,24008},X(0)}
{{99,111,187,3098,5027,6031,9999,12042,14691},X(0)}
{{4,23,111,112,148,895,10752,14671,15745},X(0)}
{{3,111,112,468,620,2030,2492,6593,11623},X(0)}
{{2,111,112,186,5166,5622,6091,14651,15565},X(0)}
{{2,107,111,125,468,1560,1637,14697,15366},X(0)}
{{16,111,187,5994,6105,6108,6137,11549},X(0)}
{{15,111,187,5995,6104,6109,6138,11537},X(0)}
{{14,111,115,5619,6109,9201,11549,11626},X(11627)}
{{13,111,115,5618,6108,9200,11537,11624},X(11625)}
{{6,111,148,827,6033,9301,9999,11641},X(0)}
{{107,111,148,12131,12384,13166,13202},X(0)}
{{25,110,111,136,858,1560,15106},X(0)}
{{3,111,352,2080,6233,7610,19911},X(0)}
{{3,111,115,858,895,3565,5107},X(0)}
{{2,111,263,1316,2395,6037,22735},X(0)}
{{2,4,111,935,11188,14833,14977},X(0)}
{{111,187,230,352,647,2715},X(0)}
{{111,186,187,468,2485,10423},X(0)}
{{111,115,265,476,13291,16188},X(0)}
{{107,111,186,1995,11643,21397},X(0)}
{{107,111,115,132,7687,11746},X(0)}
{{37,101,111,115,5513,5526},X(0)}
{{23,111,376,1296,6091,8591},X(0)}
{{23,111,323,858,5913,10420},X(0)}
{{23,111,140,620,7953,15562},X(0)}
{{23,111,137,427,933,1560},X(0)}
{{23,25,107,111,8428,13558},X(0)}
{{16,111,115,6115,10657,16806},X(0)}
{{15,111,115,6114,10658,16807},X(0)}
{{6,99,111,5971,7711,12188},X(0)}
{{6,14,15,111,5994,6772},X(11618)}
{{6,13,16,111,5995,6775},X(11617)}
{{4,111,115,468,1289,13166},X(0)}
{{3,111,353,2709,7618,14084},X(0)}
{{3,111,182,7622,13241,14684},X(0)}
{{3,4,111,5505,10097,10098},X(0)}
{{2,98,111,5912,9828,16092},X(6055)}
{{2,37,111,898,1083,11650},X(0)}
{{2,25,111,1304,2433,14685},X(0)}


Some circles on 6 or more points through X(186):
{{3,98,112,186,1691,2079,5621,14671,14675,15462},X(0)}
{{5,107,125,186,1141,5961,16337,18284,18402},X(0)}
{{3,110,186,1300,13557,14674,14703,15470,15478},X(0)}
{{2,111,112,186,5166,5622,6091,14651,15565},X(0)}
{{4,15,16,186,3484,11674,13509,15412},X(15451)}
{{3,104,108,186,1319,11700,11713,14667},X(0)}
{{112,186,378,1300,11587,15463,21397},X(0)}
{{74,112,186,376,3165,3166,15035},X(0)}
{{5,110,128,136,186,1511,2383},X(0)}
{{5,54,137,186,933,6150,11702},X(0)}
{{128,137,186,10024,13367,13557},X(0)}
{{115,186,381,14367,18402,21397},X(0)}
{{111,186,187,468,2485,10423},X(0)}
{{110,186,249,1304,7471,15468},X(0)}
{{107,186,381,13530,14644,14674},X(0)}
{{107,111,186,1995,11643,21397},X(0)}
{{36,186,242,855,1283,1459},X(0)}
{{23,186,187,8428,14729,22259},X(0)}
{{5,122,186,1301,13289,18401},X(0)}
{{5,114,186,827,3563,15462},X(0)}
{{5,112,115,186,5966,15560},X(0)}
{{4,186,265,5961,14674,22752},X(0)}
{{4,136,186,1986,6801,18402},X(0)}
{{4,108,186,915,1785,1845},X(0)}
{{4,107,186,1300,14222,21396},X(0)}
{{3,125,128,186,5667,11562},X(0)}
{{3,99,186,3563,15560,19165},X(0)}
{{3,74,107,186,11587,13558},X(0)}
{{2,182,186,14675,15563,15744},X(0)}
{{2,98,107,186,14652,23239},X(0)}
{{1,5,186,1785,11700,23961},X(0)}

Oddly, only one of those in the X(186) list has as a center in ETC.

The center of {3,98,112,..} is:

= X(3)X(2799)∩X(22)X(1637)

Barycentrics  a^2 (b^2-c^2) (a^8-a^6 b^2-a^4 b^4+a^2 b^6-a^6 c^2-a^4 b^2 c^2+2 a^2 b^4 c^2-a^4 c^4+2 a^2 b^2 c^4-2 b^4 c^4+a^2 c^6) : : 

=2 X[7663]-3 X[9175].

 

= lies on these lines: {3,2799},{22,1637},{32,2507},{523,15646},{804,12042},{3268,15246},{6636,9979},{7485,14417},{7630,8673},{7663,9175}.


The circle {6,11,37,55} has center:

a^2 (b-c) (a^9 b-3 a^8 b^2+2 a^7 b^3+2 a^6 b^4-4 a^5 b^5+4 a^4 b^6-2 a^3 b^7-2 a^2 b^8+3 a b^9-b^10+a^9 c-5 a^8 b c+9 a^7 b^2 c-10 a^6 b^3 c+9 a^5 b^4 c-4 a^4 b^5 c-a^3 b^6 c+2 a^2 b^7 c-2 a b^8 c+b^9 c-3 a^8 c^2+9 a^7 b c^2-13 a^6 b^2 c^2+16 a^5 b^3 c^2-14 a^4 b^4 c^2+5 a^3 b^5 c^2+3 a^2 b^6 c^2-6 a b^7 c^2+3 b^8 c^2+2 a^7 c^3-10 a^6 b c^3+16 a^5 b^2 c^3-6 a^4 b^3 c^3-2 a^3 b^4 c^3-4 a^2 b^5 c^3+12 a b^6 c^3-8 b^7 c^3+2 a^6 c^4+9 a^5 b c^4-14 a^4 b^2 c^4-2 a^3 b^3 c^4+10 a^2 b^4 c^4-7 a b^5 c^4+2 b^6 c^4-4 a^5 c^5-4 a^4 b c^5+5 a^3 b^2 c^5-4 a^2 b^3 c^5-7 a b^4 c^5+6 b^5 c^5+4 a^4 c^6-a^3 b c^6+3 a^2 b^2 c^6+12 a b^3 c^6+2 b^4 c^6-2 a^3 c^7+2 a^2 b c^7-6 a b^2 c^7-8 b^3 c^7-2 a^2 c^8-2 a b c^8+3 b^2 c^8+3 a c^9+b c^9-c^10):: 

= on lines: {}.

Best regards,
Peter Moses.