The isotomic disc and the isogonal disc conjecture

Let ABC be a triangle with incentre I, circumcentre O, orthocentre H, centroid G and symmedian point K. In standard notation, the triangle ABC has sides a, b, c, semiperimeter s, circumradius R and inradius r. Euler’s well-known result that the incentre I is always within the orthocentroidal disc DGH, the disc with diameter GH, is probably the first result about the location of the incentre in some disc formed by triangle centres. Investigating the location of the incentre I in other discs, in [1] we proved that the incentre is interior to the Brocard disc DOK, that is, the disc with diameter OK. The disc is named after the French military meteorologist and geometer Henri Brocard (1845-1922), known in triangle geometry for the Brocard points and the Brocard angle (see [2]).