HYACINTHOS 28545

 

[Antreas P. Hatzipolakis]:

 

 

 

Let ABC be a triangle.

 

Denote:

 

A', B', C' = the midpoints of AI, BI, CI, resp.

 

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

 

 

 

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[Ercole Suppa]

 

The reflections of Ra, Rb, Rc in AI, BI, CI, resp. are concurrent at a point: 

 

Q = X(1)X(3) ∩ X(5)X(24042) 

 

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

 

= 3*X[376]+X[10526], X[548]-X[5841], 5*X[631]-X[10525]

 

= lies on these lines: {1,3}, {5,24042}, {21,11231}, {140,3825}, {186,1872}, {355,6950}, {376,10526}, {404,11230}, {548,5841}, {631,10525}, {4188,5886}, {4302,6958}, {4996,10914}, {5267,5690}, {5428,10164}, {5440,5694}, {5657,17548}, {5881,18515}, {5887,17100}, {6684,7508}, {6713,15171}, {6833,18407}, {6842,24466}, {6882,15338}, {6905,22793}, {6906,18480}, {6914,9956}, {6924,9955}, {6935,18517}, {6942,12699}, {10572,12619}, {10993,24390}, {12515,21740}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,23961},{3,35,1385},{3,1482,7280},{3,2077,3579},{3,10679,5204},{3,10902,17502},{3,11849,36},{35,14792,3057},{36,11849,10222},{6914,25440,9956}

 

= (6-8-13) search numbers [1.19701574093541227, 1.28298377528357856, 2.19997614166401905]

 

 

Best regards

Ercole Suppa