Let $G$ be a graph of order $n \geqslant 4$. Prove that if $G$ has $t_{2}(n)+1$ edges then it contains two triangles with a common edge. Here, $t_{2}(n)=\left\lfloor n^{2} / 4\right\rfloor$ is the Turán number.

Suppose instead that $G$ has exactly one triangle. Show that $G$ has at most $t_{2}(n-1)+2$ edges, and that this number can be attained.