A circle and two semicircles are nested as shown. What’s the proportion of their respective radii a : b : c ?
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The proportion a : b : c is 1 : φ : φ2, where φ is the Golden Ratio.
The strategy for the solution is to derive three expressions for the length of line segment AO, giving two equations one can solve.
For the second expression we use the fact that |AO|=|OI|, because O is a common point of three tangents and therefore the Tangent-tangent theorem can be applied twice. Now from right triangle OKE, one can derive that |OE|2=c2-b2. Right triangle OIE then gives |AO|2=|OI|2=c2-2b2.
Now from equating c2-2ac, c2-2b2 and ab, and setting a=1, one arrives at the desired solution.