TUYMAADA 2018 S1

Do there exist three different quadratic trinomials $f(x), g(x), h(x)$ such that the roots of the equation $f(x)=g(x)$ are $1$ and $4$ , the roots of the equation $g(x)=h(x)$ are $2$ and $5$ , and the roots of the equation $h(x)=f(x)$ are $3$ and $6$ ?

Proposed by A. Golovanov