We can infer the proof from existence of isogonal conjugate too. If we are to show that X = AB ∩ DE, Y = BC ∩ EF, Z = CD ∩ FA are collinear for conconical ABCDEF, then notice that △ADY and △CYF are similar, and that X and Z will correspond to the isogonal conjugate if we overlap the similar triangles. This means that ∠DYX = ∠CYZ, hence making XYZ collinear.