We suppose that $L$ is context-free.

Let $s$ be a word of the language, $s=a^pb^pc^{2p}$, where $p$ is the pumping length.

Then from the pumping lemma, we can write $s=uvwxy$ such that $|vwx| \leq p$, $|vx| \geq 1$ and $uv^iwx^iy, \forall i \geq 0$.

There are the following cases:

1. $vwx = a^j$ for some $j \leq p$.
2. $vwx = a^jb^k$ for some $j$ and $k$ with $j+k \leq p$.
3. $vwx = b^j$ for some $j \leq p$.
4. $vwx = b^jc^k$ for some $j$ and $k$ with $j+k \leq p$.
5. $vwx = c^j$ for some $j \leq p$.