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)