(am+bn)^2+(an-bm)^2
=(am)^2+2abmn+(bn)^2+(bm)^2-2abmn+(an)^2
=(am)^2+(bn)^2+(bm)^2+(an)^2
=a^2(m^2+n^2)+b^2(m^2+n^2)
=(a^2+b^2)(m^2+n^2)