
theorem
  for a,b,c,e,f,g being POINT of TarskiEuclid2Space st
    Tn2TR a, Tn2TR b, Tn2TR c is_a_triangle &
    angle(Tn2TR a, Tn2TR b, Tn2TR c) < PI &
    angle(Tn2TR e, Tn2TR c, Tn2TR a) = angle(Tn2TR b,Tn2TR c,Tn2TR a) / 3 &
    angle(Tn2TR c, Tn2TR a, Tn2TR e) = angle(Tn2TR c, Tn2TR a, Tn2TR b) / 3 &
    angle(Tn2TR a, Tn2TR b, Tn2TR f) = angle(Tn2TR a, Tn2TR b, Tn2TR c) / 3 &
    angle(Tn2TR f, Tn2TR a, Tn2TR b) = angle(Tn2TR c, Tn2TR a, Tn2TR b) / 3 &
    angle(Tn2TR b, Tn2TR c, Tn2TR g) = angle(Tn2TR b, Tn2TR c, Tn2TR a) / 3 &
    angle(Tn2TR g, Tn2TR b, Tn2TR c) = angle(Tn2TR a, Tn2TR b, Tn2TR c) / 3
      holds
  dist(f,e) = dist(g,f) & dist(f,e) = dist(e,g) & dist(g,f) = dist(e,g)
  proof
    let a,b,c,e,f,g be POINT of TarskiEuclid2Space;
    assume
A1: Tn2TR a, Tn2TR b, Tn2TR c is_a_triangle &
    angle(Tn2TR a, Tn2TR b, Tn2TR c) < PI &
    angle(Tn2TR e, Tn2TR c, Tn2TR a) = angle(Tn2TR b,Tn2TR c,Tn2TR a) / 3 &
    angle(Tn2TR c, Tn2TR a, Tn2TR e) = angle(Tn2TR c, Tn2TR a, Tn2TR b) / 3 &
    angle(Tn2TR a, Tn2TR b, Tn2TR f) = angle(Tn2TR a, Tn2TR b, Tn2TR c) / 3 &
    angle(Tn2TR f, Tn2TR a, Tn2TR b) = angle(Tn2TR c, Tn2TR a, Tn2TR b) / 3 &
    angle(Tn2TR b,Tn2TR c, Tn2TR g) = angle(Tn2TR b, Tn2TR c, Tn2TR a) / 3 &
    angle(Tn2TR g, Tn2TR b, Tn2TR c) = angle(Tn2TR a, Tn2TR b, Tn2TR c) / 3;
    |.Tn2TR f - Tn2TR e.| = dist(f,e) & |. Tn2TR g - Tn2TR f .| = dist(g,f) &
    |.Tn2TR e - Tn2TR g.| = dist(e,g) by ThEquiv;
    hence thesis by A1,EUCLID11:23;
  end;
