In the configuration, illustrated in Figure 1, ABC is a triangle with I 1; I 2, I 3 the excentres opposite A, B, C respectively. The triangles I 1 BC, I 2 CA, I 3 AB are denoted by T 1, T 2, T 3 respectively. Ok , Hk , are the circumcentre, orthocentre respectively of triangle Tk , k = 1, 2, 3. Lk is the Euler line of Tk , k = 1, 2, 3. Define the point Ek (t) on Lk by the equation