  In this paper, we show how geometry plays in the study of the Furstenberg
conjecture (refer to~\cite{F}). Let $p>1$ and $q>1$ be two relative prime
positive integers. We prove that a non-atomic $p$- and $q$-invariant measure
having balanced geometry must be the Lebesgue measure. In the proof, we will
not assume the ergodicity of the measure. The result provides an intuitive
geometric criterion to either prove the Furstenberg conjecture or construct a
counter-example.
