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Wheels within wheels

Three circles, two of them equal, are drawn in contact with a semicircle as shown.

The radius of the semicircle is 12cm. What is the radius of the smallest circle?


The radius of the smallest circle is 3cm. Since the normal to the common tangent at S passes through the centres of the circles, SRB is a straight line. Using Pythagoras' theorem: in triangle RTP, RT2 = (a+b)2 - (a-b)2; in triangle RTB, RT2 = (2a - b)2 - b2; Equating the two expressions for RT2, gives a = 2b, so b = 3cm

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