1. In a triangle ABC of area Δ, BC = a, CA = b, AB = c are the sides; 0 the centre and R the radius of the circumcircle; ρ 1 and ρ 2 the radii of the first and second Lemoine circles; σ the radius of the Brocard circle; O 9 the centre of the nine-points circle; Ω, Ω′ the Brocard points and ω the Brocard angle; K the Lemoine point; G the centroid; H the orthocentre; m a , m b , m c the medians; h a , h b , h c the altitudes; AL = s a , BS = s b , CT = s c the symmedians corresponding to the sides a, b, c; 2∝ and 2β the axes of the Brocard ellipse of foci Ω, Ω′ and centre V; 2x and 2y the axes of the Lemoine ellipse of foci G, K and centre ϕ τ and τ′ the Torricelli segments (the invariant segments which join a vertex of ABC to the vertices of the equilateral triangles constructed on the opposite side, internally and externally to the fundamental triangle); p(X ) the power of a point X with respect to the circle O(R ).