On a tiling problem of R.B. Eggleton

A tiling of the plane with polygonal tiles is said to be strict if any point common tcl two tiles is a vertex of both or a vertex of neither. A triangle is said tc> be rational if its sides have rational length. Recently R.B. Eggleton asked if it is pos+le to strictly tile the plane <with rational triangles us@ preci.sely one triangle from each congruence clas\. In this paper we cDnstructivelv prove the existence of such a tiling by a suitable modification of the technique !qgested by E_g@et#n. The theory of rational points on elliptic curves, in parricular,‘the Nageil-Lutz theorem. plays a crucial role in completing the proof. .