Prove that curve with zero torsion is planar

I have proved that a planar curve of zero curvature is a straight line. It follows from the Frenet equations.
But now I need to prove that if $\varkappa=0$, then the space curve $\mathbf{r}(t)$ is planar.
From the condition and the Frenet equations it follows that
$$ \left\{
\right. $$

But how can be technically deduced from these equations that the curve is planar?

Update: from a related question planar curve if and only if torsion I have realized that I need to show that $(\mathbf{r}(t)-\mathbf{r}(t_0))\cdot\mathbf{b}(t)=0$ for any $t$ and some $t_0$. The question now is how to do that.
I appreciate any help.

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