HYACINTHOS 27555

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, IaIbIc the antipedal tringle of I and Fe, Fa, Fb, Fc  the Feuerbach point and the three ex-Feuerbach points.

The circumcircles of FeAIa, FeBIb, FeCIc are coaxial.
Let F0 be the second, other than Fe, intersection.

 

The circumcircles of FaAIa, FaBIb, FaCIc are coaxial.
Let F1 be the second, other than Fa, intersection.

The circumcircles of FbAIa, FbBIb, FbCIc are coaxial.
Let F2 be the second, other than Fb, intersection.

The circumcircles of FcAIa, FcBIb, FcCIc are coaxial.
Let F3 be the second, other than Fc, intersection.

The points F0, F1, F2, F3 are concyclic.

Center of the circle?

[Angel Montesdeoca]:

The points F0=X(5531), F1, F2, F3 lies on circle of center:

W = (r+3R) X(1) - 4R X(5) =  X(1)X(5) /\ X(40)X(64)

with first barycentric coordinate:
 
 a (-a^6+2 a^5 (b+c)+a^4 (b^2+c^2)-2 a^3 (2 b^3+b^2 c+b c^2+2 c^3)+a^2 (b+c)^2 (b^2+c^2)+2 a (b^5-b^3 c^2-b^2 c^3+c^5)-(b-c)^2 (b+c)^4)

 
W is the reflection of X(i) in X(j), for these {i, j}: {40,5687}, {46,11499}, {1898,5777}, {6264,12740}, {10073,119}, {10085,3}, {12704,3149}.

W lies on lines X(i)X(j) for these {i, j}:  {1,5}, {3,210}, {4,2900}, {8,6261}, {9,10902}, {10,6889}, {20,4420}, {33,2910}, {35,7330}, {40,64}, {46,912}, {55,1898}, {63,6796}, {78,515}, {84,1259}, {100,1158}, {101,7079}, {224,5552}, {254,1826}, {354,6918}, {404,12757}, {411,3681}, {474,12675}, {480,971}, {518,3149}, {581,612}, {610,2917}, {936,958}, {944,997}, {946,3870}, {962,3935}, {990,4878}, {1071,1376}, {1260,12664}, {1709,11248}, {1745,8270}, {1750,5812}, {1766,3949}, {1858,11501}, {1867,3191}, {2948,12890}, {3085,10393}, {3158,12705}, {3174,11372}, {3185,15623}, {3295,9848}, {3306,12005}, {3338,6911}, {3359,15071}, {3434,12608}, {3475,6864}, {3586,10953}, {3689,10306}, {3694,5776}, {3711,5584}, {3873,6915}, {3880,6765}, {3913,12672}, {3940,14110}, {4294,5811}, {4413,9940}, {4557,15622}, {4853,11014}, {4855,5450}, {5119,5887}, {5178,6932}, {5220,5732}, {5248,15064}, {5268,5788}, {5440,12114}, {5657,12520}, {5691,10526}, {5709,5904}, {5779,15837}, {5791,8580}, {5794,15843}, {5818,6877}, {5927,11496}, {6245,6745}, {6256,10522}, {6259,11826}, {6282,11827}, {6684,10884}, {6826,13407}, {6834,10916}, {7702,12831}, {8583,16842}, {10175,10585}, {10830,15177}, {11012,12687}, {11491,12514}, {12407,13214}, {12775,13548}, {16139,16143}.

 (6 - 9 - 13) - search numbers  of W: (-4.20253732826351, -8.71491257242104, 11.6136981066283).
 
 Angel Montesdeoca