HYACINTHOS 28549

 

[Antreas P. Hatzipolakis]:

 

Variation 2:

 

Let ABC be a triangle and A'B'C' the antipedal triangle of I.

 

Denote:

 

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)

 

Point of concurrence of Ra, Rb, Rc (radical center of the circles) =: U.

The reflections of Ra, Rb, Rc in AI, BI, CI, resp. are concurrent at a point W such that W and U are symmetric in I.

 

 

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[Ercole Suppa]

 

 

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

 

U = MIDPOINT OF X(6583) AND X(12672)

 

= -a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-4 a^4 b c+9 a^3 b^2 c+6 a^2 b^3 c-10 a b^4 c-2 b^5 c-a^4 c^2+9 a^3 b c^2-18 a^2 b^2 c^2+9 a b^3 c^2+b^4 c^2-2 a^3 c^3+6 a^2 b c^3+9 a b^2 c^3+4 b^3 c^3+2 a^2 c^4-10 a b c^4+b^2 c^4+a c^5-2 b c^5-c^6)  : : (barys)

 

= 7*X[1385]-3*X[5918], 3*X[1699]+X[10284], X[2771]-X[3881], 9*X[3656]-X[3868], 5*X[3890]+3*X[12699], 5*X[5439]-3*X[13145], 3*X[5603]-X[5885], X[6583]+X[12672] 

 

= lies on these lines: 1,22461}, {5,10}, {149,18480}, {1385,5918}, {1621,13624}, {1699,10284}, {2771,3881}, {3585,9957}, {3656,3868}, {3890,12699}, {5180,12600}, {5439,13145}, { 5603,5885}, {6264,10222}, {6583, 12672}, {15178,18444}, {16160, 21630}

 

= midpoint X(6583) and X(12672)

 

= (6-8-13) search numbers [-2.33951669983917064, -2.70642741715640644, 6.59412193986458539]

 

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

 

W = MIDPOINT OF X(944) AND X(5885)  

 

= a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c+12 a^4 b c-7 a^3 b^2 c-10 a^2 b^3 c+6 a b^4 c-2 b^5 c-a^4 c^2-7 a^3 b c^2+14 a^2 b^2 c^2-7 a b^3 c^2+b^4 c^2-2 a^3 c^3-10 a^2 b c^3-7 a b^2 c^3+4 b^3 c^3+2 a^2 c^4+6 a b c^4+b^2 c^4+a c^5-2 b c^5-c^6)  : : (barys)

 

= X[517]-X[550], X[944]+X[5885], X[2771]-X[3884], 9*X[3655]-X[3869], 5*X[3889]+3*X[18481], 2*X[4540]-3*X[11812]

 

= lies on these lines: {1,22461}, {517,550}, {944,5885}, {1385,5251}, {2771,3884}, { 2975,4420}, {3655,3869}, {3889,18481}, {4540,11812}, {4857, 5049}, {5045,5434}, {6224,10914}, {6912,15178}

 

= midpoint X(944) and X(5885)

 

= (6-8-13) search numbers [5.72013371875323695, 6.08704443607047275, -3.21350492095051908]

 

 

Best regards

Ercole Suppa