A theorem on concurrent Euler lines

Abstract

In the configuration, illustrated in Figure 1, ABC is a triangle with I1; I2, I3 the excentres opposite A, B, C respectively. The triangles I1BC, I2CA, I3AB are denoted by T1, T2, T3 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