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A circle of radius $r$ is drawn inside a triangle so that it
just touches each of the three sides as shown in the diagram. The
three corners and points where the circle touches have been
labelled $A$ to $F$.
One side of the triangle is divided into segments of length
$a$ and $b$ by the inscribed circle. However, we are not told which
of the three sides is divided in this way.
From this information we can find an expression for the area
of the triangle. Prove that the area of the triangle is:
$$\frac{abr(a+b)}{abr^2}. $$

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.
If a is the radius of the axle, b the radius of each ballbearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.