Is This Proof Number 368?
Seattle-area mathematics Teacher D.E. and his student C.J. recently produced a novel proof of the Pythagorean Theorem using two concentric circles. In their words, the proof was "quite by accident... arrived on our mental doorstep in class."
- BC is tangent to circle A at C
- XW is a line through C
- BV is tangent to circle A at C
Prove: AC2+CB2 = AB2
Hint: Think chord relationships!
Solution Commentary: D.E. explains: "By labeling, AB = c, BC = a, AC = b. c is the radius of the big circle, b is the radius of the small circle. By the intersecting chords theorem, (XC)(CW) = (BC)(CV).
Algebraically, (c+b)(c-b) = (a)(a).
Expand the binomial and complete."
After congratulating D.E., the student and possibly yourself for producing a successful, clever proof, the question is: Is this a new proof of the Pythagorean Theorem?
Unfortunately, it is a "known" proof. In fact, it is Proof #58 in Elisha Loomis' The Pythagorean Proposition (1940), which is a text compilation of all "known" proofs of the Pythagorean Theorem (up to 367 different proofs presently!).
A version of this proof first appeared in George Edwards' Elements of Geometry (1895).
Using the on-line copy (or your own if you have it!), see Problem #102 on page 154, associated with Figure 9 on page 156. The diagram/set-up and proof is exactly the same (except you do not see the inner circle).
The proof also supposedly appeared in the Journal of Education in 1887, but a search of the full year on-line found no record of it...
Thanks D.E. and his student for an enjoyable find!