continuous partial derivatives with constant over circles are not zero

Question

Let $f:\mathbb R^2 \to \mathbb R$ with continuous partial derivatives such that $f(C_i),$ for $i=1,2,3$ is constant over $C_i$ being a circle of radius $1$ with centers at $(-1,0), (1,0), (0,\sqrt 3$).

Then exists a point $p \in \bigcup_i C_i $ where $\nabla f = 0 $

I have no idea where to start, the only thing I noticed is that the circles touch in $3$ points.

Answer 1

Hint If $f$ is constant on a circle $C$, it means that $\nabla f$ is orthogonal to the circle at every point of $C$. If $\nabla f$ is never $0$, it means that it is constantly directed outwards, or constantly inwards the circle.