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Mind your decisions, Im Presh Talwalkar. In semicircle A, construct semicircles B and C along the diameter of semicircle A. Then construct circle D which is internally tangent to semicircle A and externally tangent to semicircles B and C. If the radius of semicircle B is equal to 2 and the radius of semicircle C is equal to 1, solve for the radius of circle D. This problem appeared on the 2017 AMC 12a exam. Pause the video, if youd like to give this problem a try and when youre ready, keep watching to learn how to solve this problem. We begin with a preliminary result. If two circles are tangent, their centers and the tangent points are collinear. To see why, draw a tangent line through Point C. By definition, AC will meet this at a right angle and also BC will meet this tangent line at a right angle. Therefore, ACB is a straight line and all points A, C, and B are along the same line. This is true for externally tangent circles and its also true for internally tangent circles. R