HYACINTHOS 27134

[ Tran Quang Hung]:

Let ABC be a triangle. Two Fermat points F1 and F2.

A1B1C1 and A2B2C2 is cevian triangle of F1 and F2.

S is symmedian point of ABC.

Then Isogonal conjugate of S wrt A1B1C1 and A2B2C2 are on Euler line of ABC.

Which are these points ?

[César Lozada]:

 

S1 = (S^2+sqrt(3)*(SA-3*SW+8*R^2)*S-3*(SB+SC)*R^2+SA^2+SB*SC-SW^2)*( sqrt(3)*SB*SC+S*SA) : : (barys)

= on line {2,3}

= [ 0.2217797095115145, -0.6483252738596672, 3.9871451901896040 ]

 

S2= (S^2-sqrt(3)*(SA-3*SW+8*R^2)*S-3*(SB+SC)*R^2+SA^2+SB*SC-SW^2)*(sqrt(3)*SB*SC-S*SA) : : (barys)

= on line {2, 3}

= [ -234.3331980936569000, -234.5845411231475000, 274.1991305334666000 ]

 
Midpoint of S1 and S2 was checked, but no relations with ETC centers was found.

 
César Lozada