Here is a problem related to yours and solved before.
This is good news; a problem for which the solution is 
known and which is connected with our present problem
is certainly welcome. It is still more welcome if the
connection is close and the solution simple. There is a good
chance that such a problem will be useful in solving our 
present one.
     The situation that we are discussing here is typical and
important. In order to see it clearly let us compare it
with the situation in which we find ourselves when we 
are working at an auxiliary problem.  In both cases, our 
aim is to solve a certain problem A and we introduce and 
consider another problem B in the hope that we may 
derive some profit for the solution of the proposed problem
A from the consideration of that other problem B.
The difference is in our relation to B. Here, we succeeded
in recollecting an old problem B of which we
know the solution but we do not know yet how to use it.
There, we succeeded in inventing a new problem B; we 
know(or at least we suspect strongly) how to use B, but
we do not know yet how to solve it. Our difficulty 
concerning B makes all the difference between the two
situations. When this difficulty is overcome, we may use B in
the same way in both cases; we may use the result or the 
method (as explained in AUXILIARY PROBLEM, 3), and, if
we are lucky, we may use both the result and the method.
In the situation considered here, we know well the solution
of B but we do not know yet how to use it. Therefore
we ask: Could you use it? Could you use it's result? 
Could you use its method?
     The intention of using a certain formerly solved problem
influences our conception of the present problem.
Trying to link up the two problems, the new and the
old, we introduce into the new problem elements corresponding
to certain important elements of the old problem.
For example, our problem is to determine the
sphere circumscribed about a given tetrahedron. This is
a problem of solid geometry. We may remember that we
have solved before the analogous problem of plane
geometry of constructing the circle circumscribed about
a given triangle. Then we recollect that in the old problem
of plane geometry, we used the perpendicular bisectors
of the sides of the triangle. It is reasonable to try
to introduce something analogous into our present
problem. Thus, we may be led to introduce into our present
problem, as corresponding auxiliary elements, the
perpendicular bisecting planes of the edges of the tetrahedron.
After this idea, we can easily work out the
solution to the problem of solid geometry, following the 
analogous solution in plane geometry.
     The foregoing example is typical. The consideration
of a formerly solved related problem leads us to the 
introduction of auxiliary elements, and the introduction
of suitable auxiliary elements makes it possible for us to
use the related problem to full advantage in solving our
present problem. We aim at such an effect when, thinking 
about the possible use of a formerly solved related
problem, we ask: Should you introduce some auxiliary
element in order to make its use possible?
     Here is a theorem related to ours and proved before
 This version of the remark discussed here is exemplified in
section 19.