  In this paper, we are dealing with the solution of the functional equation $$
\varphi\Big(\frac{x+y}2\Big)(f(x)-f(y))=F(x)-F(y), $$ concerning the unknown
functions $\varphi,f$ and $F$ defined on a same open subinterval of the reals.
Improving the previous results related to this topic, we describe the solution
triplets $(\varphi,f,F)$ assuming only the continuity of $\varphi$. As an
application, under natural conditions, we also solve the equality problem of
two-variable Cauchy means and two-variable quasi-arithmetic means.
