Exercise 3.5b of do Carmo's *Riemannian Geometry* asks the reader to prove that given a Killing field $X$ on a manifold $M$, an isolated zero $p$ of $X$, and a normal neighborhood $U$ of $p$ in which $X$ has no other zeros, $X$ is tangent (in $U$) to the geodesic spheres centered at $p$.

I have a "proof", but it doesn't use the fact that $X\_p=0$, so I must be missing something. Here's the "proof":

Let $\phi\_t$ be the flow of $X$ on $U$, and let $q=\exp\_p{v}\in U$. Since $\phi\_t$ is an isometry, $\phi\_t(q) = \exp\_{\phi\_t(p)} d(\phi\_t)\_p v$. So, denoting the radial distance function at $p$ by $r\_p$, we get
$$r\_{\phi\_t(p)}(\phi\_t(q)) = \|d(\phi\_t)\_pv\| = \|v\|,$$
which means $\frac{d}{dt}\!\left[r\_{\phi\_t(p)}(\phi\_t(q)) \right]=0$. But by the chain rule,
$$\frac{d}{dt}\!\left[r\_{\phi\_t(p)}(\phi\_t(q)) \right] = 
\begin{pmatrix}
\text{stuff} &d(r\_p)\_q X\_q
\end{pmatrix},$$
where $\text{stuff}$ is an unimportant block matrix. Thus, $d(r\_p)\_q X\_q=0$. Hence, $\left\langle \frac{\partial}{\partial r\_p}, X\right\rangle\equiv 0$. Thus, $X$ is tangent to the geodesic spheres in $U$.

What did I miss?