Vol. 7 No 5 - Pi Mu Epsilon

I
Let CG = a,GF= b,CF=CA=CD= r,CE=x,ED= y,AE=u, andEF= v.
Then we have
orean, then DE/EC is rational.
from similar triangles AEC and AFG.
Then
so DE/EC is rational when a, b, and r are integral, establishing the
converse.
If DE/EC = p/q is rational, with p even, take m = q + p/2 and
2 2 2
n = p/2 and let a' = 2tm, b' = m - n , and P' = m2 + n . Then we have
+ at - bt- m2 t n2 + 2mn - m 2 + n - 2 n - p
bt m2 - n2 m- n q y
but this Pythagorean triangle with sides a', b', and r' is similar to
triangle CFG.
It is easy to find counterexamples showing that the stated theorem
is quite false.
Counterexamples were submitted by RICHARD A. GIBBS, BOB PRIELIPP,
and ROBERT A.
STUMP.
Editor's comient.
When confronted with her mistake, the proposer
pwred that "after all, rational and integral are almost the same thing."
Such pussy-footing evades the problem, but Zelda is a regular contributor
of high quality, so this is no catastrophe.
477. [Fall 19801 Pftopo.~ed by Solomon W. Gotomb, Univ&u^ty 06
Southem Cati