# $3$ mutually tangent circles

$\textbf{Problem}$: In $\triangle{ABC}$, $AB=3$, $BC=4$, and $CA=5$. Additionally, we have mutually tangent circles $X$, $Y$, and $Z$ inside the triangle that are tangent to $\{AB, BC\}$, $\{BC, CA\}$, and $\{CA, AB\}$ respectively. Determine the sum of the radii of the circles $X$, $Y$, and $Z$.

$\textbf{Thoughts}$: I drew a diagram. However, I do not know asymptote. I'm aware that drawing a diagram is best in these types of geometry problems, but my diagram (which was grotesque) did not help me.

An addendum by Jack: the above diagram depicts Steiner's construction of the Malfatti circles of a $3-4-5$ triangle; the blue lines are the bitangents mentioned by the linked Wikipedia article.

Let $a$, $b$ and $c$ be radius of circles.
Thus, we need to solve the following system. $$a\sqrt5+2\sqrt{ab}+b=3,$$ $$3c+2\sqrt{bc}+b=4,$$ $$a\sqrt5+2\sqrt{ac}+3c=5.$$