HYACINTHOS 27556

[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.

1. Point F0 ?

2. Center of the circle?

 

 

[Peter Moses]:

Hi Antreas,

1). X(5531).

2).
Radius of circle = 2 R^2 / (R - r).

Center of circle = 

a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+2 a^3 b^2 c-2 a^2 b^3 c+2 b^5 c-a^4 c^2+2 a^3 b c^2-2 a^2 b^2 c^2+2 a b^3 c^2-b^4 c^2+4 a^3 c^3-2 a^2 b c^3+2 a b^2 c^3-4 b^3 c^3-a^2 c^4-b^2 c^4-2 a c^5+2 b c^5+c^6):: 

 

2 X[1837] - 3 X[5587], 4 X[496] - 5 X[8227], 3 X[5660] - X[12750].
(3 R + r) X[1] - 4 R X[5].
(2 R + r) X[3] - 3 R X[210].

 

= on the cubic K364 and these lines: {{1,5},{3,210},{4,2900},{8,626 1},{9,10902},{10,6889},{20,442 0},{33,2910},{35,7330},{40,64} ,{46,912},{55,1898},{63,6796}, {78,515},{84,1259},{100,1158}, {101,7079},{224,5552},{254,182 6},{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},{176 6,3949},{1858,11501},{1867,319 1},{2948,12890},{3085,10393},{ 3158,12705},{3174,11372},{3185 ,15623},{3295,9848},{3306,1200 5},{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},{12115,17647},{12332, 17661},{12407,13214},{12775, 13548},{16139,16143}}.

 

= 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}}.

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5531, 5534), (12, 355, 5587), (72, 11500, 40), (100, 12528, 1158), (200, 1490, 40), (355, 10942, 10827), (3689, 12688, 10306), (3913, 12672, 12703), (5534, 5720, 1), (5881, 6326, 1).

 

Best regards,
Peter Moses.