  Let $P,Q$ be longest paths in a simple graph. We analyze the possible
connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and
introduce the notion of a bi-traceable graph. We use the results for all the
possible configurations of the intersection points when $\#V(P)\cap V(Q)\le 5$
in order to prove that if the intersection of three longest paths $P,Q,R$ is
empty, then $\#(V(P)\cap V(Q))\ge 6$. We also prove Hippchen's conjecture for
$k\le 6$: If a graph $G$ is $k$-connected for $k\le 6$, and $P$ and $Q$ are
longest paths in $G$, then $\#(V(P)\cap V(Q))\ge 6$.
