HYACINTHOS 28556

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

 

Denote:

 

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

 

A'B'C' = the reflection triangle of NaNbNc

(ie A', B', C' = the reflections of Na, Nb, Nc in NbNc, NcNa, NaNb, resp.)

 

(Oa), (Ob), (Oc) = the circles (A', A'Na), (B', B'Nb), (C', C'Nc), resp.

 

Ra = the radical axis of (Ob), (Oc)

Rb = the radical axis of (Oc), (Oa)

Rc = the radical axis of (Oa), (Ob)

 

The reflections of Ra, Rb, Rc in AI, BI, CI, resp. are concurrent (on the OI line).

 

APH

 

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[Ercole Suppa]

 

 

*** Ra, Rb, Rc are concurrent at a point U

 

U = MIDPOINT OF X(21) AND X(16138)

 

= a (2 a^6-a^5 b-5 a^4 b^2+2 a^3 b^3+4 a^2 b^4-a b^5-b^6-a^5 c+6 a^4 b c+a^3 b^2 c-2 a^2 b^3 c-4 b^5 c-5 a^4 c^2+a^3 b c^2+2 a^2 b^2 c^2+a b^3 c^2+b^4 c^2+2 a^3 c^3-2 a^2 b c^3+a b^2 c^3+8 b^3 c^3+4 a^2 c^4+b^2 c^4-a c^5-4 b c^5-c^6)  : : (barys)

 

= X[10]-X[30], 3*X[191]-X[12702], 3*X[381]-X[16118], X[517]-X[3652], X[758]-X[11278], 3*X[1699]-X[16150], 4*X[3634]-3*X[5499], X[3648]+X[12699], 3*X[5886]-X[16116], X[8148]+3*X[13465], 6*X[10021]-5*X[19862], 4*X[12104]-3*X[17502], 3*X[15677]-X[18481], 3*X[16159]-X[20084], 3*X[16160]-2*X[18483]

 

= lies on these lines: {1,399}, {3,5506}, {10,30}, {21,4881}, {79,3582}, {191,12702}, {355,13199}, {381,16118}, {517,3652}, {758,11278}, {942,16141}, {1012,5694}, {1699,16150}, {2475,7705}, {3634,5499}, {3648,12699}, {5428,15254}, {5659,16113}, {5885,6912}, {5886,16116}, {6841,7173}, {6888,16128}, {6906,22935}, {7743,16153}, {8148,13465}, {9780,18516}, {9957,16140}, {10021,19862}, {10225,19925}, {11230,12608}, {11263,12611}, {12104,17502}, {15677,18481}, {16117,18540}, {16159,20084}, {16160,18483}

 

= midpoint of X(i) and X(j) for these {i,j}: {21,16138}, {7701,13743}

 

= reflection of X(i) in X(j) for these {i,j}: {79,9955}, {3579,3647}, {18480,22798}, {22937,22936}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3579,3647,22937}, {3579,22936,3647}

 

= (6-8-13) search numbers [-1.48719801507818180, -2.78978417177342497, 6.25845260778668030]

 

 

*** The reflections of Ra, Rb, Rc in AI, BI, CI, resp. are concurrent at X(3579)

 

 

Best regards

Ercole Suppa