HYACINTHOS 26115

[Alexander Skutin (solver6)] (*)

 

 

1. Let ABC be a triangle and A'B'C' the pedal triangle of O.

Denote:

Hbc, Hcb = the orthocenters of IBC', ICB', resp.
Hca, Hac = the orthocenters of ICA', IAC', resp.

Hab, Hba = the orthocenters of IAB', IBA', resp.

A*B*C* = the triangle bounded by HbcHcb, HcaHac, HabHba

The circumcircle of A*B*C* passes through the Feuerbach point.

Which point is its center?

 

 

2. Let ABC be a triangle. 

Denote:

Ab, Ac = the orthogonal projections of A on BI, CI, resp.
Bc, Ba = the orthogonal projections of B on CI, AI, resp.
Ca, Cb = the orthogonal projections of C on AI, BI, resp.

Hbc, Hcb = the orthocenters of IBAc, ICAb, resp.
Hca, Hac = the orthocenters of ICBa, IABc, resp.

Hab, Hba = the orthocenters of IACb, IBCa, resp.

A*B*C* = the triangle bounded by HbcHcb, HcaHac, HabHba

The circumcircle of A*B*C* passes through the Feuerbach point.

Which point is its center?

 

(*)
https://artofproblemsolving.co m/community/c6h1455586p8391375


[Angel Montesdeoca]:

 

 

*** 1.   Cicumcenter of  A*B*C* is

W = (2 a^3 (b+c)+a^2 (b^2-8 b c+c^2)-2 a (b^3-2 b^2 c-2 b c^2+c^3)-(b^2-c^2)^2: ... : ...)

W is the midpoint of X(i) and X(j) for these {i,j}: {4,10912}, {962,12513}, {1320,13271}, {3161,12512}.

W is  reflection of X(i) in X(j) for these {i,j}:  {8715,5901}, {10915,9955}, {12607,946}.

W lies on lines X(i)X(j) for these {i,j}:  {1, 528}, {4, 10912}, {5, 2802}, {8, 10896}, {10, 3829}, {11, 8256}, {12, 3885}, {145, 5229}, {149, 10950}, {355, 5854}, {392, 9710}, {404, 13205}, {496, 6797}, {516, 11260}, {517, 3813}, {518, 4301},    {519, 3845}, {529, 12699}, {908, 3893}, {946, 3880}, {962, 12513}, {1001, 9785}, {1145, 7741}, {1320, 10944,}, {1329, 10914}, {1537, 5881}, {1697, 6690}, {1699, 3680}, ....

with (6-9-13)-search number (1.64100784459787,-4.11073638897120,5.72917066907312).


**** 2. The triangle A * B * C * of cases 1. and 2. coincide.

Angel Montesdeoca