J.A. Garfield (Garfield, 183 1 ~ 1888), the 7th president, had a keen interest and superb talent in elementary mathematics when he was a student. 1876, (then a congressman, elected president of the United States five years later) gave a beautiful proof of Pythagorean theorem, which was published in the New England Journal of Education. The idea of proof is to use the area formulas of trapezoid and right triangle. As shown on the next page, it is a right-angled trapezoid composed of three right-angled triangles. Find the same area with different formulas

that is

a2+2ab+b2=2ab+c2

a2+b2=c2

This kind of proof is often of interest to middle school students when they study geometry.