I got the impression that you were quoting or paraphrasing an old comment from someone else, too. I thought it was interesting.

It's true the proof doesn't use identities (and how could it, most of them take advantage of the pythagorean theorem at some point), but it does use basic trig relationships--similar triangles--even if we were to sidestep the definition of cosine.

The discussion has been along the lines of what constitutes a trig proof, as opposed to algebraic or geometric--since they're pretty inter-related and each can be expressed in terms of the other, more or less. Now, a "graphical" proof is probably not too ambiguous, it's usually "I know it when I see it."

Anyway, what are the chances of adding it to the list? :)

Yes, Loomis mentions this in his book.

>It's true the proof doesn't use identities (and how could
>it, most of them take advantage of the pythagorean theorem
>at some point), but it does use basic trig
>relationships--similar triangles--even if we were to
>sidestep the definition of cosine.

So, this is not what the discussion is about. OK.
>
>The discussion has been along the lines of what constitutes
>a trig proof, as opposed to algebraic or geometric--since
>they're pretty inter-related and each can be expressed in
>terms of the other, more or less. Now, a "graphical" proof
>is probably not too ambiguous, it's usually "I know it when
>I see it."

In my view, a proof deserves to be referred to as "trigonometric" if without trigonometry it makes no sense. I do not believe your derivation passes this test. Instead of using cosine, state simply the proportions that follow from the similarity of triangles. That the ratios involved got honored with a notation used widely is irrelevant to the proof.

Another question to ask is what constitutes a distinct proof.

>Anyway, what are the chances of adding it to the list? :)

If you refer to my list, zero.

I'm definitely not claiming it is a distinct proof. Most of the proofs are interesting variations, though.

I think it is most nearly the same as entry #6, but by using the cosine definition, it becomes more succinct.

>If you refer to my list, zero.

Maybe a note added to #6? :)

It does acquire a distinct flavor. I can't make my mind about the taste, though.

>Maybe a note added to #6? :)

This would make sense, yes. Thank you.

>>I think it is most nearly the same as entry #6, but by using
>>the cosine definition, it becomes more succinct.
>
>It does acquire a distinct flavor. I can't make my mind
>about the taste, though.
>
>>Maybe a note added to #6? :)
>
>This would make sense, yes. Thank you.

In this particular case of a proof of the Pythagorean theorem, a simple proof based on the similarity of triangles is quite self-contained. Introducing cosine into the proof does not even simplify it. cosine just embodies the fact of similarity and nothing less; its use modifies the verbiage of the proof. There is nothing trigonometric about this change.