The triangle ABC has sides AB = 137, AC = 241, and BC =200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of CD. [asy][asy] pair A = (2,6); pair B = (0,0); pair C = (10,0); pair D = (3.5,0) ; pair E = (3.1,2); draw(A--B); draw(B--C); draw(C--A); draw (A--D); dot ((3.1,1.7)); label ("E", E, dir(45)); label ("A", A, dir(45)); label ("B", B, dir(45)); label ("C", C, dir(45)); label ("D", D, dir(45)); draw(circle((1.8,1.3),1.3)); draw(circle((4.9,1.7),1.75)); [/asy][/asy]