Main Concept

Let AB, the diameter of a semicircle, be divided at point C so that $\frac{\mathrm{AC}}{\mathrm{CB}} \= k$. If DE is the diameter of the circle inscribed in the arbelos which is tangent to all three semicircles, then $\frac{\mathrm{DE}}{\mathrm{AB}} \= \frac{k}{k^2\+k\+1}$ .

AB = 2

AC =

CB =

DE =

 k = $\frac{\mathrm{AC}}{\mathrm{CB}}$=

Therefore:

$\frac{k}{k^2\+k\+1}$ =

$\frac{\mathrm{DE}}{\mathrm{AB}}$ =

ratio $\frac{\mathrm{AC}}{\mathrm{AB}}$ =

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MathApps/AlgebraAndGeometry