  This article discusses a geometric perspective on the well-known fact in
graph theory that the effective resistance is a metric on the nodes of a graph.
The classical proofs of this fact make use of ideas from electrical circuits or
random walks; here we describe an alternative approach which combines geometric
(using simplices) and algebraic (using the Schur complement) ideas. These
perspectives are unified in a matrix identity of Miroslav Fiedler, which
beautifully summarizes a number of related ideas at the intersection of graphs,
Laplacian matrices and simplices, with the metric property of the effective
resistance as a prominent consequence.
