Triangle $$ABC$$ is a right triangle with $$AC = 7,$$ $$BC = 24,$$ and right angle at $$C.$$ Point $$M$$ is the midpoint of $$AB,$$ and $$D$$ is on the same side of line $$AB$$ as $$C$$ so that $$AD = BD = 15.$$ Given that the area of triangle $$CDM$$ may be expressed as $$\frac {m\sqrt {n}}{p},$$ where $$m,$$ $$n,$$ and $$p$$ are positive integers, $$m$$ and $$p$$ are relatively prime, and $$n$$ is not divisible by the square of any prime, find $$m + n + p.$$

(第二十一届AIME2 2003 第11题)