  Kemeny's constant is an invariant of discrete-time Markov chains, equal to
the expected number of steps between two states sampled from the stationary
distribution. It appears in applications as a concise characterization of the
mixing properties of a Markov chain and has many alternative definitions. In
this short article, we derive a new geometric expression for Kemeny's constant,
which involves the distance between two points in a simplex associated to the
Markov chain: the circumcenter and the Lemoine point. Our proof uses an
expression due to Wang, Dubbeldam and Van Mieghem of Kemeny's constant in terms
of effective resistances and Fiedler's interpretation of effective resistances
as edge lengths of a simplex.
