HYACINTHOS 29112

[Antreas P. Hatzipolakis]:

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

Denote:

A", B", C" = the reflections of I in BC, CA, AB, resp.

A*, B*, C* = the reflections of I in AA", BB", CC", resp.

La, Lb, Lc = The reflections of AA*, BB*, CC* in BC, CA, AB, resp.

A1, B1, C1 = points on IA', IB', IC', resp. such that A1I/A1A' = B1I/B1B' = C1I/C1C'  = t

1. La, Lb, Lc are concurrent. 

Point?

 

2. the parallels to La, Lb, Lc through A1, B1, C1, resp. are concurrent.

Locus of the point of concurrence as t varies?

[César Lozada]:

 

1)      Q1 = X(35)X(37) ∩ X(65)X(16118)

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

= lies on these lines: {35, 37}, {65, 16118}, {5663, 5903}

= {X(267), X(1717)}-harmonic conjugate of X(35)

= [ -5.3174748888193530, -5.6475963570457490, 10.0047580623942100 ]

 

2)      Q2(t) lies on the line through ETCs {1, 3024, 3028, 5663, 7727, 19470, 32143, 32168, 33535}.

 

César Lozada