2001 Pan African MO Problems/Problem 6
Let be a semicircle with centre and diameter .A circle with centre is drawn, tangent to , and tangent to at . A semicircle is drawn, with centre on , tangent to and to . A circle with centre is drawn, internally tangent to and externally tangent to and . Prove that is a rectangle.
We use Cartesian coordinates, setting , and assume without loss of generality that is closer to than to and that the -coordinate of is positive. Then we compute (because that is the only choice that allows for tangency to both and the line ). Letting be the radius of , we find that . However, since and are tangent, we know that ; using the Pythagorean theorem, we then solve , which solves to , so that .
Finally, we know that must now be uniquely determined. If was a rectangle, then would have to be located at , so we only need check that a circle centered there is tangent to . Letting denote the radius of , we know from tangencies that we must have . These equations are all satisfied for our desired choice of and the value , so we conclude that is rectangular and we are done.
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