NCTM Calendar Problem (11-2-2016)

Creation of this applet was inspired by the the calendar problem (11/2/2016) that appeared in NCTM's November 2016 issue of the Mathematics Teacher magazine. "In right triangle ABC, CD is the bisector of right angle ACB, CM is the median to the hypotenuse, and CP is the altitude to the hypotenuse. Prove CD bisects angle PCM. " This applet informally illustrates what this calendar problem is asking you to prove. Can you formally prove what this applet informally illustrates?