$O_1$ and $O_2$ are centers of $\odot$$(ABE)$ and $\odot$$(BCD)$,respectively.$B$$O_1$$\cap$$\odot$$(ABE)$=$B_1$.$B$$O_2$$\cap$$\odot$$(BCD)$=$C_1$.$B_1$$A$$\perp$$AB$,$C_1$$D$$\perp$$AB$,$C_1$$C$$\perp$$BC$,$BC$$\perp$$B_1$$E$ $\Rightarrow$ $AB_1$$\parallel$$DC_1$,$B_1M$$\parallel$$C_1C$ $\Rightarrow$ $\angle$$B_1AM$$=$$\angle$$C_1MC$,$\angle$$C_1CM=$$\angle$$AMB_1$ $\Rightarrow$ $\bigtriangleup$$AB_1M$$\equiv$$\bigtriangleup$$MC_1C$ $\Rightarrow$ $B_1M=C_1C$, hence $B_1MCC_1$ is a parallelogram.Hence $B_1C_1=MC=$$\frac{AC}{2}$ and $O_1O_2=$$\frac{AC}{4}$,since $O_1$ and $O_2$ are midpoints of $BB_1$ and $BC_1$,respectively.