HYACINTHOS 29365

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle, L the Euler line and A'B'C', A"B"C" the pedal triangles of H, O, resp. (orthic, medial triangles, resp.)

A*, B*, C* = the orthogonal projections of A', B', C' on L, resp.

A**, B**, C** = the orthogonal projections of A", B", C" on A'A*, B'B*, C'C*, resp.

The centroid of A**B**C** lies on the Euler line.

[Peter Moses]:

Hi Antreas,

>The centroid of A**B**C** lies on the Euler line.

= 2*a^10*b^2 - 5*a^8*b^4 + 3*a^6*b^6 + a^4*b^8 - a^2*b^10 + 2*a^10*c^2 + 2*a^8*b^2*c^2 - a^6*b^4*c^2 - 7*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - b^10*c^2 - 5*a^8*c^4 - a^6*b^2*c^4 + 12*a^4*b^4*c^4 - 4*a^2*b^6*c^4 + 4*b^8*c^4 + 3*a^6*c^6 - 7*a^4*b^2*c^6 - 4*a^2*b^4*c^6 - 6*b^6*c^6 + a^4*c^8 + 5*a^2*b^2*c^8 + 4*b^4*c^8 - a^2*c^10 - b^2*c^10 : :

= lies on these lines: {2,3}, {51,523}, {275,1304}, {1624,8901}, {1994,14611}, {2452,9777}, {2453,17810}, {2790,12099}, {5097,30221}, {6795,10601}, {10412,14583}, {12079,13567}, {16319,23292}

Best regards,
Peter Moses.