Just touching
Three semi-circles have a common diameter, each touches the other
two and two lie inside the biggest one. What is the radius of the
circle that touches all three semi-circles?
Age
16 to 18
Challenge level
Problem
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Two semi-circles are drawn on one side of a line segment. Another semi-circle touches them externally as shown in the diagram.
What is the radius of the circle that touches all three semi-circles in terms of the radii of the first two semi-circles?
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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
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Let the large semi-cirle have diameter $AB$ and centre $X$.
Let the two smaller semi-circles 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
Simplifying gives:
There is a pleasing symmetry about this formula which gives the radius of the small circle in terms of the radii of the two smaller semicircles. An excellent solution, well done Sue!
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