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# geometry

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AB is a line segment and M is its midpoint. Semicircles are drawn with AB , AM and MB as diameters on the same side of line AB. A circle with center O is drawn which touches the three semicircles, then the radius of the smallest cicle is equal to x/y*AB​.  Find x/y.

Jun 23, 2020

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AB is a line segment and M is its midpoint.
Semicircles are drawn with AB , AM and MB as diameters on the same side of line AB.
A circle with center O is drawn which touches the three semicircles,
then the radius of the smallest circle is equal to $$\dfrac{x}{y}*AB$$.
Find $$\dfrac{x}{y}$$.

$$\text{Let the radius of the smallest circle =r }\\ \text{Let R=\dfrac{AB}{4} }$$

$$\begin{array}{|rcll|} \hline (\mathbf{R+r)^2} &=& \mathbf{R^2+(2R-r)^2} \\ R^2+2Rr+r^2 &=& R^2+ 4R^2-4Rr+r^2 \\ 2Rr &=& 4R^2-4Rr \\ 2Rr+4Rr &=& 4R^2 \\ 6Rr &=& 4R^2 \quad | \quad :6R \\\\ r &=& \dfrac{4}{6}*R \\\\ r &=& \dfrac{2}{3}*R \quad | \quad R=\dfrac{AB}{4} \\\\ r &=& \dfrac{2}{3}*\dfrac{AB}{4} \\\\ r &=& \dfrac{1}{3}*\dfrac{AB}{2} \\\\ \mathbf{r} &=& \mathbf{\dfrac{1}{6}*AB} \\ \hline \end{array}$$

$$\mathbf{\dfrac{x}{y}=\dfrac{1}{6}}$$

Jun 23, 2020