Geometry brain warmup for high school students

The standard American high school math curriculum seems to regard geometry as something to be gulped down as quickly as possible, rather than savored and thought about in order to develop geometric intuition. Many bright high school students take geometry in 8th or 9th grade and their teachers then believe they are "done" with geometry.

Award-winning math teacher Pat Ballew, who taught in schools in Japan and England for 16 years, has a different perspective.

Mr. Ballew posted the following:

In the January, 1929, issue of the American Mathematical Monthly, there appears a problem submitted by J. Rosenbaum of Milford Connecticut. The problem begins, "It is well known that the radius of the inscribed circle of a right triangle is equal to half the difference between the sums of the legs and the hypotenuse." I ... suggest that the theorem suggested may be less well known now than it might have been in the past.

If it's been a while since you've taken geometry, and you're feeling uncomfortably rusty, see if you can prove this so-called "well-known theorem" by playing around with it, then look at Mr. Ballew's very nice presentation and discussion of several approaches to proving this theorem. Even if you came up with your own proof, you may find you get new insights by looking at his approaches.

If you liked working on this problem, there are more where that one came from.