Just touching
Problem
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Two semicircles are drawn on one side of a line segment. Another semicircle touches them externally as shown in the diagram.
What is the radius of the circle that touches all three semicircles in terms of the radii of the first two semicircles?

Getting Started
Find lengths in terms of the radii of the two red semicircles.
Student Solutions
This solution comes from Sue Liu of Madras College
Let the large semicirle have diameter $AB$ and centre $X$.
Let the two smaller semicircles have centres $C$ and $D$ and radii $R$ and $r$.
Thus $AC = R$, $BD = r$ so that $AX = (2R + 2r)/2 = (R + r)$ and $CX = r$ from which it follows that $XD = R$.
Let the small circle have have centre $O$ and radius $x$.
Then $CO = R + x$ and $DO = r + x$.
The line $XO$, joining the centre of the large semicircle to the centre of the small circle, cuts the circumference of the large semicircle at $E$ where $XE = XB = R + r$, $OE = x$ and $OX = R + r  x$.
If we now consider the triangle $OCD$ we have
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$\angle OXC + \angle OXD = 180^o$
So
\begin{eqnarray}\frac{r^2 + (R + r  x)^2  (R + x)^2}{2r(R + r  x)} + \frac{R^2 + (R + r  x)^2  (r + x)^2}{2R(R + r  x)} = 0\end{eqnarray}
Teachers' Resources