  We present a formula that expresses the Hankel determinants of a linear
combination of length $d+1$ of moments of orthogonal polynomials in terms of a
$d\times d$ determinant of the orthogonal polynomials. This formula exists
somehow hidden in the folklore of the theory of orthogonal polynomials but
deserves to be better known, and be presented correctly and with full proof. We
present four fundamentally different proofs, one that uses classical formulae
from the theory of orthogonal polynomials, one that uses a vanishing argument
and is due to Elouafi [J. Math. Anal. Appl. 431} (2015), 1253-1274] (but given
in an incomplete form there), one that is inspired by random matrix theory and
is due to Br\'ezin and Hikami [Comm. Math. Phys. 214 (2000), 111-135], and one
that uses (Dodgson) condensation. We give two applications of the formula. In
the first application, we explain how to compute such Hankel determinants in a
singular case. The second application concerns the linear recurrence of such
Hankel determinants for a certain class of moments that covers numerous
classical combinatorial sequences, including Catalan numbers, Motzkin numbers,
central binomial coefficients, central trinomial coefficients, central Delannoy
numbers, Schr\"oder numbers, Riordan numbers, and Fine numbers.
