Mechanics — Euler's theorem on homogeneous parts of $T$

A particle of mass $m$ moves on a horizontal $xy$-plane rotating about the vertical $z$-axis with angular velocity $\omega$. If $T=T_2+T_1+T_0$ with $T_2=\frac12m(\dot x^2+\dot y^2)$, $T_1=m\omega(x\dot y-y\dot x)$ and $T_0=\frac12m\omega^2(x^2+y^2)$, then $\dot x\frac{\partial T}{\partial\dot x}+\dot y\frac{\partial T}{\partial\dot y}$ is equal to: