Let ABC be a triangle, F be the first (or secon) Fermat point, let K be the point on the Kiepert hyperbola
Let P be the point on line FK. The line through P and perpendicular to BC meet AP at A_0. Define A_0,B_0,C_0
cyclically. Show that A_0B_0C_0 is an equilateral triangle. This triangle homothety to the outer(or inner) Napoleon triangle