Proof confusion or solving strategy?

Question

I found myself reading a proof to $Claim X$ for which the proof involved the use of the Truth of $Claim Y$, later I found a proof to $Claim Y$ (from a different source) and was suprised to find that it involved the use of the Truth of $Claim X$. Of course I suspected/knew that there must be other ways to prove atleast one of them, but as from my perspective it was a unique situation. (The Claims in question was "Any polygon can be triangulated" and "The sum of all interior angles in a polygon is $180(n-2)$ degrees") When i from my perspective overlooked the situation it seemed like none of either $Claim X$ or $Y$ have been proved as of yet, only that

Truth of $Claim X$ and reasoning led to Truth of $Claim Y$,

Truth of $Claim Y$ and reasoning led to Truth of $Claim X$,

i.e. Truth of $Claim X$ $\Leftrightarrow$ Truth of $Claim Y$

Questions: Do there exist problems which require you to create such a scenario as stated above and utilize it as a problemsolving strategy somehow OR is the situation just weird and should be avoided?

and

Given an equivalence between two claims such that "if $ClaimA$ is True then so is $ClaimB$", aswell as "if $ClaimB$ is True then so is $ClaimA$". If there is a way to prove $A$ directly which implies $B$, is the opposit always possible aswell, that is, is there then always a way to directly prove $B$ which implies $A$?

Also, the comments and answers below might be of help understanding what this is all about. Thanks in advance.

Answer 1

I think you are correct in suspecting that this equivalence can be a problem solving technique, especially when tackling very difficult problems.

As a simple example of this, suppose that you are attempting to find all rational solutions of a family of biquadratic equations and suppose that, in the standard way, you have established an equivalence with rational points on a family of elliptic curves. As part of your problem solving strategy you can now switch backwards and forwards between the two scenarios depending upon which seems more tractable for the problem in hand.

From what I have read, the proof of F.L.T. was a splendid example of switching between different viewpoints according to the difficulties that were arising in each possible approach.

Answer 2

I think I understand your question.

In the context you describe, each of theorems $X$ and $Y$ implies the other. Each seems to be of about the same level of difficulty. You know two sources, one of which proves $X$ first, using some geometric arguments, then deduces $Y$ essentially as a corollary. The other source does the reverse.

This kind of situation is common, and not at all weird. It is often instructive to provide both approaches, to make it clear that $X$ and $Y$ are (in some sense) at the same level of difficulty.

One classic example is Euclid's parallel postulate. Any of the following (and many more) could be taken as an axiom - then the others would be theorems.

• Given a line and a point not on it there is just one line through the point parallel to the line.
• The sum of the angles of a triangle is two right angles.
• There are similar triangles that are not congruent.