Circles $C_1$ and $C_2$ each have radius $5$, and the distance between their centers is $4$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$?

1. Let $O_1=(-2,0)$ and $O_2=(2,0)$ be the centers of $C_1,C_2$. By symmetry $C_3$ and $C_4$ are centered on the $y$-axis. A circle centered at $(0,k)$ with radius $\rho$ that is internally tangent to $C_1$ (inside it) satisfies $\sqrt{2^2+k^2}=5-\rho$.
2. The radius is largest when $k=0$, so $C_3$ is centered at the origin with radius $r_3=5-2=3$.
3. For $C_4$ at $(0,k)$ with radius $r_4$: $\sqrt{4+k^2}=5-r_4$, and, since $C_4$ is externally tangent to $C_3$, $k=3+r_4$.
4. Substituting: $(5-r_4)^2=4+(3+r_4)^2$, i.e. $25-10r_4=13+6r_4$, so $16r_4=12$ and $r_4=\boxed{\dfrac34}$.