HYACINTHOS 29309

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle 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

 

The circumcircles of IA'A*, IB'B*, IC'C* are coaxial.

 

Locus of the 2nd (other than I) intersection as t varies?

 

 

[Ercole Suppa]:

 

 

The locus of the 2nd (other than I) intersection as t varies is: 

 

Γ:  ∑ b (a-b-c)^3 (b-c) c x^3 + 4 a (a-b) b (a^3-a^2 b-a b^2+b^3+5 a b c) x y z -a (a-c) (a+b-c) (2 a-2 b+c) (a-b+c)^2 y^2 z+a (a-b) (2 a+b-2 c) (a+b-c)^2 (a-b+c) y z^2 = 0     (barys)

 

*** Centers X(i) ∈ Γ for these i: {1,515,1317,1319,1323,11700,15524,15730}

 

 

*** Let P = P(t) be the 2nd (other than I) intersection of circumcircles of IA'A*, IB'B*, IC'C*. 

 

Pairs (t,P(t) = X(i)) with t = number: {(t = 0,X(1319)), (t = 1,X(515)), (t = 2,X(15524))}

 

 

*** Some other points: 

 

P(1/2) = X(1)X(1389) ∩ X(5844)X(11696)

 

= a (a^2-b^2+3 b c-c^2) (2 a^7-7 a^6 b+2 a^5 b^2+13 a^4 b^3-10 a^3 b^4-5 a^2 b^5+6 a b^6-b^7-7 a^6 c+26 a^5 b c-22 a^4 b^2 c-13 a^3 b^3 c+28 a^2 b^4 c-13 a b^5 c+b^6 c+2 a^5 c^2-22 a^4 b c^2+46 a^3 b^2 c^2-23 a^2 b^3 c^2-6 a b^4 c^2+3 b^5 c^2+13 a^4 c^3-13 a^3 b c^3-23 a^2 b^2 c^3+26 a b^3 c^3-3 b^4 c^3-10 a^3 c^4+28 a^2 b c^4-6 a b^2 c^4-3 b^3 c^4-5 a^2 c^5-13 a b c^5+3 b^2 c^5+6 a c^6+b c^6-c^7) : : (barys)

 

= lies on these lines: {1,1389}, {5844,11696}

 

= (6-9-13)  search numbers:  [-1.4686866585809874226, -1.6154267753699688062, 5.4368922457394286255]

 

--------------------------------------------------------------------------

 

P(2/3) = X(1)X(227) ∩ X(1319)X(5577)

 

= a (a^2-b^2+4 b c-c^2) (2 a^7-9 a^6 b+4 a^5 b^2+17 a^4 b^3-14 a^3 b^4-7 a^2 b^5+8 a b^6-b^7-9 a^6 c+40 a^5 b c-33 a^4 b^2 c-20 a^3 b^3 c+41 a^2 b^4 c-20 a b^5 c+b^6 c+4 a^5 c^2-33 a^4 b c^2+68 a^3 b^2 c^2-34 a^2 b^3 c^2-8 a b^4 c^2+3 b^5 c^2+17 a^4 c^3-20 a^3 b c^3-34 a^2 b^2 c^3+40 a b^3 c^3-3 b^4 c^3-14 a^3 c^4+41 a^2 b c^4-8 a b^2 c^4-3 b^3 c^4-7 a^2 c^5-20 a b c^5+3 b^2 c^5+8 a c^6+b c^6-c^7) : : (barys)

 

= lies on these lines: {1,227}, {1319,5577}

 

= (6-9-13)  search numbers:  [-2.5640964124146839562, -2.8895240622671632846, 6.8245333307453462799]

 

--------------------------------------------------------------------------

 

P(3/2) = X(1)X(10308) ∩ X(484)X(5951)

 

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

 

= lies on these lines: {1,10308}, {484,5951}, {1319,3024}, {2605,14838}

 

= (6-9-13)  search numbers:  [3.0748524143612344832, 3.2826186943093039661, -0.0510803438580164811]

 

 

Best regards,

Ercole Suppa