smc

Question

In the figure below, semicircles with centers at A and B and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter JK. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at P is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at P? FIGURE

Accepted Answer

Connect the centers of the tangent circles! (call the center of the large circle C) FIGURE Notice that we don't even need the circles anymore; thus, draw triangle ABP with cevian PC: FIGURE and use Stewart's Theorem: AB AC BC + AB CP^2 = AC BP^2 + BC AP^2 From what we learned from the tangent circles, we have AB = 3, AC = 1, BC = 2, AP = 2 + r, BP = 1 + r, and CP = 3 - r, where r is the radius of the circle centered at P that we seek. Thus: 3 1 2 + 3 (3-r)^2 = 1 (1+r)^2 + 2 (2+r)^2 6 + 3(9 - 6r + r^2) = (1 + 2r + r^2) + 2(4 + 4r + r^2) 33 - 18r + 3r^2 = 9 + 10r + 3r^2 28r = 24 r = 67 67