On a property of 2-connected graphs and Dirac's Theorem

We refine a property of $2$-connected graphs described in the classical paper of Dirac from 1952 and use the refined property to somewhat shorten Dirac's proof of the fact that each $2$-connected $n$-vertex graph with minimum degree at least $k$ has a cycle of length at least $\min\{n,2k\}$.