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 D GH , 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 D OK , 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]).