Here is a problem from the homework of Math Challenge III class.  There are multiple solutions, and everyone is encouraged to find your own solution.  I will post solutions later.

In right triangle $ABC$, $\angle C=90^{\circ}$, $AB=13$, $AC=12$.  Circles $\odot O_1$ and $\odot O_2$ both have radius $r$, and $\odot O_1$ and $\odot O_2$ are externally tangent.  Also given that $\odot O_1$ is tangent to $\overline{AB}$ and $\overline{AC}$ at $M$ and $G$ respectively, $\odot O_2$ is tangent to $\overline{AB}$ and $\overline{BC}$ at $N$ and $H$ respectively.  Find the value of $r$.