HYACINTHOS 24975

[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.

L1, L2, L3 = the Euler lines of AIA", BIB", CIC", resp.

A*B*C* = the triangle bounded by L1, L2, L3

1. A'B'C', A*B*C* are orthologic

2. A"B"C", A*B*C* are orthologic

[Peter Moses]:

Hi Antreas,

 

1) (r + R) X[1] + (r + 4 R) X[7] = 3 a^3 b+a^2 b^2-3 a b^3-b^4+3 a^3 c+4 a^2 b c+3 a b^2 c+a^2 c^2+3 a b c^2+2 b^2 c^2-3 a c^3-c^4:: {{1,7},{10,3901},{11,113},...} .

the other orthology X(1621).

 

2) 3 R X[1] - 2(r + 4 R) X[7] = a^4+2 a^3 b-2 a b^3-b^4+2 a^3 c+a^2 b c+2 a b^2 c+2 a b c^2+2 b^2 c^2-2 a c^3-c^4:: {{1,7},{4,5561},{30,5425},...} .

the other orthology, also X(1621).

 

Best regards,

Peter Moses.