HYACINTHOS 28380

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, A'B'C' the pedal triangle of O and A"B"C" the pedal triangle of I.

 

Denote:

 

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

 

A*, B*, C* = points on A'Na, B'Nb, C'Nc such that:

 

A'A* / A'Na = B'B* / B'Nb = C'C* / C'Nc = t

 

D = the Poncelet point of ABCI = Feuerbach point X(11)

 

La =: DA*,  Lb =: DB*, Lc =: DC*

 

L1, L2, L3 = the reflections of La, Lb, Lc in AI, BI, CI, resp.

 

1. L1, L2, L3 are concurrent

 

Which is the locus of the point of concurrence as t varies?

 

2. The parallels to L1, L2, L3 through A", B", C", resp. are concurrent.

 

Which is the locus of the point of concurrence as t varies?

 

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[Ercole Suppa]

 

(1) locus1 = rectangular hyperbola through ETC points: X(65), X(513), X(517), X(1317), centered at X(5083)

 

(2) locus2 = rectangular hyperbola through ETC points: X(65), X(513), X(517), X(13755), X(13756) 

centered at 

 

P1 =  

 

= MIDPOINT OF X(65) AND X(13756)

 

Barycentrics  -a (a^7 (b+c)+6 a^5 b c (b+c)-2 a^6 (b+c)^2-(b-c)^6 (b+c)^2+2 a^2 b (b-c)^2 c (3 b^2-2 b c+3 c^2)+a^4 (3 b^4-6 b^3 c-2 b^2 c^2-6 b c^3+3 c^4) +a^3 (-3 b^5+3 b^4 c+b^3 c^2+b^2 c^3+3 b c^4-3 c^5)+a (b-c)^2 (2 b^5-6 b^4 c+3 b^3 c^2+3 b^2 c^3-6 b c^4+2 c^5)) : :  

 

= X[65]+X[13756], 3*X[354]-X[3025], X[14115]-2*X[18240]

= lies on these lines:  {1,953},{57,901},{59,1421},{65,13756},{226,3259},{354,3025},{513,5083},{517,1387},{1319,18593},{2835,15635},{5570,12016},{11028,18839},{14115,18240}

 

= midpoint of X(65) and X(13756)

 

= reflection of X(14115) in X(18240)

 

= 6-9-13 search numbers [0.285869856852859396, 0.978460897766639955, 2.83132854182923172]

 

Best regards

Ercole Suppa