HYACINTHOS 28554

[Antreas P. Hatzipolakis]:

 

 

Variation 3:

 

Let ABC be a triangle.

 

Denote:

 

A', B', C' = the circumcenters of IBC, ICA, IAB, 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)

 

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(4) AND X(10284)

 

= 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+7 a^3 b^2 c+4 a^2 b^3 c-8 a b^4 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+4 a^2 b c^3+7 a b^2 c^3+2 a^2 c^4-8 a b c^4+b^2 c^4+a c^5-c^6)  : : (barys)

 

= X[4]+X[10284], X[546]-X[2802], X[550]-3*X[3898], 3*X[1482]+X[5693], X[2771]-X[7984], X[2800]-X[6583], X[5694]+X[7982], X[5885]-2*X[13464], X[5887]+X[11278], 2*X[5901]-X[13145], 5*X[11522]-X[25413]

 

= lies on these lines: {4,10284}, {5,10}, {65,16173}, {392,17531}, {546,2802}, {550,3898}, {962,6951}, {1385,6909}, {1482,5693}, {2771,7984}, {2800,6583}, {3057,3585}, {3579,6940}, {5441,5919}, {5603,6972}, {5694,7982}, {5697,10895}, {5885,13464}, {5887,11278}, {5901,13145}, {6284,9957}, {10058,24928}, {10738,12751}, {11009,17638}, {11522,25413}, {15558,18990}, {18393,25414}

 

= midpoint of X(i) and X(j) for these {i,j}: {4,10284}, {3057,22793}, {5887,11278}, {10222,12672}, {18480,23340}

 

= reflection of X(i) in X(j) for these {i,j}: {5885,13464}, {13145,5901}

 

= (6-8-13) search numbers [-2.68587794894358968, -2.99242695479864457, 6.95198065781894365]

 

 

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

 

W =  MIDPOINT OF X(550) AND X(3874)

 

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

 

= 5*X[3]-X[5904], X[30]-X[6583], X[140]-X[2801], X[382]-5*X[18398], X[389]-2*X[15229], X[515]-X[5885], X[517]-X[550], X[518]-X[14810], 5*X[632]-3*X[15064], X[912]-X[12038], X[952]-X[13145], X[971]-X[9955], 2*X[3530]-X[3678], 3*X[3576]-X[5694], X[3579]-3*X[10167], 3*X[3656]+X[9961], X[6001]-X[15178], X[6102]+X[23156], 3*X[7967]-X[10284], 3*X[10202]+X[12680], 3*X[10246]+X[15071], 3*X[11220]+X[12699], 3*X[11231]-X[14872], 5*X[15016]-X[18525]

 

= lies on these lines: {3,5904}, {21,104}, {30,6583}, {35,17660}, {65,4325}, {72,5303}, {79,354}, {140,2801}, {382,18398}, {389,15229}, {515,5885}, {517,550}, {518,14810}, {632,15064}, {912,12038}, {942,7354}, {946,12267}, {952,13145}, {971,9955}, {1858,5126}, {3057,11571}, {3530,3678}, {3576,5694}, {3579,10167}, {3583,13751}, {3656,9961}, {5045,10391}, {5083,15171}, {5536,16117}, {5563,17637}, {6001,15178}, {6102,23156}, {6940,12738}, {7967,10284}, {8582,8728}, {10202,12680}, {10225,11491}, {10246,15071}, {10268,24645}, {11220,12699}, {11231,14872}, {15016,18525}, {15931,22937}, {16132,22765}

 

= midpoint of X(i) and X(j) for these {i,j}: {550,3874}, {6102,23156}, {12675,13369}, {12680,18480}

 

= reflection of X(i) in X(j) for these {i,j}: {389,15229}, {3678,3530}, {6583,12005}, {9955,13373}, {9956,9940}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {10202,12680,18480}

 

= (6-8-13) search numbers [6.06649496785765599, 6.37304397371271088, -3.57136363890487734]

 

 

Best regards

Ercole Suppa