Derive Keplers 3rd law

Equate gravitational force (GMm/r^2) to centripetal force (mv^2/r). Rearrange to get v^2=GM/r. Due to the approximated circular orbit v=2pir/T so v^2=4pi^2r^2/T^2. Therefore GM/r=4pi^2r^2/T^2. Rearranging gives GM=4pi^2r^3/T^2. Rearranging futher gets T^2=(4pi^2/GM)r^3. Hence T^2 is proportional to r^3.