Let $\Gamma$ be a graph with $2n$ vertices and $n^2+1$ edges. Than there is two subgraphs $K', K'' \subset \Gamma$ both isomorphic to $K_3$ such that they have a common edge.
I have tried to count subgraphs isomorphic to $K_3$ using same methods as when we are proving Turan's theorem and than apply pigeonhole principle but i'am stuck...
Generalize the statement to apply to graphs with an odd number of vertices, rather than requiring $|V|=2n$, and then induct on $|V|$.
By Mantel's theorem, if the graph has at least $\frac{|V|^2}{4}+1$ edges, then it contains a triangle subgraph $K'$. If there is another triangle $K''$ sharing an edge with $K'$, then we are done. If not, then you can show that deleting the vertices of $K'$ creates a graph with $|V|-3$ vertices and at least $\frac{(|V|-3)^2}{4}+1$ edges, provided $|V| \ge 5$, and you can find what you want in the remainder by the inductive hypothesis.