HYACINTHOS 29291

[Antreas P. Hatzipolakis]:

 

 

 

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

Denote:

Oa, Ob, Oc = the circumcenters of IBC, ICA, IAB, resp.

La, Lb, Lc = the Euler lines of IBC, ICA, IAB, resp.

Kab, Kac = the perpendiculars from Oa to Lb, Lc, resp.
Kbc, Kba = the perpendiculars from Ob to Lc, La, resp.
Kca, Kcb = the perpendiculars from Oc to La, Lb, resp.

A** = Kbc /\ Kcb
B** = Kca /\ Kac
C** = Kab /\ Kba

1. ABC, A**B**C** are orthologic. 

 Orthologic centers?

2. A'B'C', A**B**C** are orthologic.
Orthologic center (A**B**C**, A'B'C') = X(21)
The other one?

3. A*B*C*, A**B**C** are orthologic.
Orthologic centers ?

Where A*B*C* is the triangle in Hyacinthos 29284  

 

 

 

[Ercole Suppa]:

 

1.  

 

Orthologic center(ABC,A**B**C**) = orthologic center(A**B**C**,ABC) =

 

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

 

= (6-9-13)  search numbers: [2.3195596563160698061, 1.9798215672093767984, 1.1994527863089321783]

 

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2. 

 

Orthologic center (A'B'C',A**B**C**) = 

 

X(1)X(22461) ∩ X(7)X(10266) =

 

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

 

=  3*X[354]-2*X[18244]  

 

= lies on these lines: {1,22461}, {7,10266}, {30,13375}, {65,17643}, {145,12849}, {224,12524}, {354,18244}, {518,12682}, {3174,12660}, {3467,22936}, {3649,12267}, {5586,12409}, {7701,12957}, {10044,13128}, {10052,13129}, {10427,12639}, {10940,13130}, {10941,13131}, {11570,12433}, {13089,14882}, {15481,18259}

 

= reflection of X(10266) in X(12917)

 

= (6-9-13)  search numbers:  [1.3555142915658257479, 1.3250316824549508386, 2.0977128748698012543]

 

Orthologic center (A**B**C**,A'B'C') = X(21)

 

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3.  

 

Orthologic center (A*B*C*,A**B**C**) = 

 

X(30)X(13100) ∩ X(2077)X(3651) =

 

= a (a^9-2 a^8 b-2 a^7 b^2+6 a^6 b^3-6 a^4 b^5+2 a^3 b^6+2 a^2 b^7-a b^8-2 a^8 c+6 a^7 b c+2 a^6 b^2 c-11 a^5 b^3 c-a^4 b^4 c+4 a^3 b^5 c+4 a^2 b^6 c+a b^7 c-3 b^8 c-2 a^7 c^2+2 a^6 b c^2+19 a^5 b^2 c^2-3 a^4 b^3 c^2-12 a^3 b^4 c^2-2 a^2 b^5 c^2-5 a b^6 c^2+3 b^7 c^2+6 a^6 c^3-11 a^5 b c^3-3 a^4 b^2 c^3+4 a^3 b^3 c^3-4 a^2 b^4 c^3-a b^5 c^3+9 b^6 c^3-a^4 b c^4-12 a^3 b^2 c^4-4 a^2 b^3 c^4+12 a b^4 c^4-9 b^5 c^4-6 a^4 c^5+4 a^3 b c^5-2 a^2 b^2 c^5-a b^3 c^5-9 b^4 c^5+2 a^3 c^6+4 a^2 b c^6-5 a b^2 c^6+9 b^3 c^6+2 a^2 c^7+a b c^7+3 b^2 c^7-a c^8-3 b c^8) : : (barys)

 

= lies on these lines: {30,13100}, {2077,3651}, {5441,10106}, {6920,7701}, {9856,15178}, {12255,13743}, {12556,17768}, {16116,22782}

 

= (6-9-13)  search numbers:  [6.7644141743361001630, 6.8886460236426736783, -4.2504354610772104417]

 

Orthologic center (A**B**C**,A*B*C*) = X(21)

 

 

Best regards,

Ercole Suppa