The two circles below are externally tangent. A common external tangent intersects line $PQ$ at $R.$ Find $QR.$ [asy] unitsize(0.3 cm); pair A, B, C, D, E; A = (0,0); B = (8,0); C = (20,0); D = intersectionpoint(Circle(A,5), arc((A + C)/2, abs(A - C)/2, 0, 180)); E = intersectionpoint(Circle(B,3), arc((B + C)/2, abs(B - C)/2, 0, 180)); draw(Circle(A,5)); draw(Circle(B,3)); draw(A--D); draw(B--E); dot("$P$", A, SW); dot("$Q$", B, S); label("$8$", (B + E)/2, W); label("$10$", (A + D)/2, W); [/asy]
