Equalities between greatest common divisors involving three coprime pairs

A new equality of the greatest common divisor (gcd) of quantities involving three coprime pairs is proven in this note. For $a_i$ and $b_i$ positive integers such that gcd$(a_i, b+i) = 1$ for $i ∈ {1, 2, 3}$ and $d_{ij} = |aibj − ajbi|$, then gcd$(d_{32}; d_{31})$ = gcd$(d_{32}; d_{21}$) = gcd$(d_{31}; d_{21})$: The proof uses properties of Farey sequences.