Students will learn what circumcircles and incircles are. Ask your students why the center of any rectangle is also the center of its circumcircle. Then ask them why the hypotenuse of any right triangle is also a diameter of its circumcircle, and further, the circumcenter is always the midpoint of the hypotenuse. This can be proven by copying the right triangle, rotating it \(180^\circ,\) and stitching it to the original right triangle, and thus forming a rectangle. Because the center of any rectangle is also the center of its circumcircle, we know the circumcenter must lie on the hypotenuse, exactly at the midpoint.
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Conclude by giving your students these challenges:
MATHCOUNTS: Find all positive integers \(n\) for which \(n^2 + 45\) is a perfect square. Here's the solution.