Home > Problem of the Week > Archive List > Detail

<< Prev 4/28/2013 Next >>

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."


  1. BC is tangent to circle A at C
  2. XW is a line through C
  3. 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!