  We show that the zeros of consecutive orthogonal polynomials $p_n$ and
$p_{n-1}$ are linearly connected by a doubly stochastic matrix for which the
entries are explicitly computed in terms of Christoffel numbers. We give
similar results for the zeros of $p_n$ and the associated polynomial
$p_{n-1}^{(1)}$ and for the zeros of the polynomial obtained by deleting the
$k$th row and column $(1 \leq k \leq n)$ in the corresponding Jacobi matrix.
