25 points ! circle b is internally tangent to circle a at point p. likewise, circle c is internally tangent to circle d at point p. if bd=ac , ab=4, and sd=2, find the length of diameters for circles c and d. be sure to justify your work. the picture is the diagram

All the given points are collinear, so we have ...
  AB +BC = AC
  BC +CS +SD = BD
The problem statement tells us
  BD = AC
  BC +CS +SD = AB +BC . . . . . substituting for BD and AC
  BC +CS +2 = 4 +BC . . . . . . . . substituting given lengths
  CS = 2 . . . . . . . . . . . . . . . . . . . subtract (2+BC)

CS is the radius of Circle C, so its diameter is twice that: 4. The radius of Circle D is the diameter of Circle C plus SD, so is 4 +2 = 6. The diameter of Circle D is twice that: 12.

The diameter of Circle C is 4.
The diameter of Circle D is 12.