On Lemoine’s Ellipse

Extract

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; ρ₁ and ρ₂ the radii of the first and second Lemoine circles; σ the radius of the Brocard circle; O₉ 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 ϕτ 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).