  Let (a,b,c)be a Pythagorean triple with c being the hypotenuse length, and h
being the altitude to the hypotenuse. Also, let v,k,l be positive integers with
k and l being relatively prime.We say(Definition1 in this work)that the
Pythagorean triple (a,b,c) has the reciprocal property R(v,k,l)if the positive
integers a,b,c,v,k,and l satisfy the condition or equation, 1/a+1/b+v/h = k/l.
The motivating force behind this work, is a problem that appeared in the
journal, Crux Mathematicorum with Mathematical Mayhem. The said problem is
Mayhem problem M390, and it appeared in the April2009 issue of the journal(see
reference {1}). A solution to the same problem was published in the February
2010 issue(see {2}).Using the above definition, the probem can be stated as
follows:Find all the Pythagorean triples that have the reciprocal property
R(1,1,1). It turns out that the only such triple is (3,4,5).Our solution to
this problem is found in Theorem2, part(iii).The results of this paper are
expressed in Theorems2,3,4 and 5. Theorem2 contains five parts.Part(ii) states
that there exists no Pythagorean triple which has the reciprocal property
R(2,k,1). Part(iv) says that if k is at least2; then there exists no
Pythagorean triple that has the reciprocal property R(1,k,l). Theorem3 states
that if the positive integer v is such that the integers v-1 and v+1 are twin
primes; then there is no Pythagorean triple that has the reciprocal property
R(v,k,1).
