Find circumradius of $\Delta DEC$

Question

$A(0,0),B(4,0)$ and $C(5,-2\sqrt 6)$ are the vertices of $\Delta ABC$. Incircle of the triangle touches side $AC$ and $BC$ at $D$ and $E$ respectively. Find the circumradius of the triangle $DEC$. Is there a short way using analytical geometry? Calculating the coordinates of $D$ and $E$ using section formula gives quite nasty coordinates. What do I do? Thanks.