Prove that a self-complementary graph has radius 2 and diameter 2 or 3.

I think that is one of the well-known properties of self-complementary graphs, but I am having some troubles trying to prove it. The facts that, if $G\cong \overline{G}$, then both graphs G and $\overline{G}$are connected, and that, for any graph, if $rad(G)\ge 3$ then $rad(\overline{G})\le 2$, and that if $diam(G)\ge 3$ then $diam(\overline{G})\le 3$, should make the proof quite easier, but I don't know how to develop it. Could you help me? Thanks in advance!

I think that is one of the well-known properties of self-complementary graphs, but I am having some troubles trying to prove it. The facts that, if $G\cong \overline{G}$, then both graphs G and $\overline{G}$are connected, and that, for any graph, if $rad(G)\ge 3$ then $rad(\overline{G})\le 2$, and that if $diam(G)\ge 3$ then $diam(\overline{G})\le 3$, should make the proof quite easier, but I don't know how to develop it. Could you help me? Thanks in advance!