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In this paper we provide a new parametrization for the diophantine equation$A^2+B^2+C^2=3D^2$ and give a series of corollaries. We discuss someconnections with Lagrange's four-square theorem. As applications, we find newparameterizations of equilateral triangles and regular tetrahedrons havinginteger coordinates in three dimensions.

New parametrization of $A^2+B^2+C^2=3D^2$ and Lagrange's four-square theorem