Using the trig functions in a proof of a geometric theorem actually means using theorems (and axioms) on similarity and congruence, so trig functions actually do not bring anything new to geometry. The term "trigonometric" has of course a historical origin, as a strict mathematical definition of trig functions comes from analysis (not necessarily by infinite series, there is a definition by inversion of algebraic integrals).

I think that in mathematics this is a wrong question. There is a multitude of sentences that could be derived from each other. There is no point in asking which came first, except historically or chronologically. But this is not what we are talking about here.

>Using the trig functions in a proof of a geometric theorem
>actually means using theorems (and axioms) on similarity and
>congruence, so trig functions actually do not bring anything
>new to geometry.

I believe I out forth an explanation as to why I think that proof desrves to be called "trigonometric". It is based on some identities that, although could be written in terms of similarities, gain at least in clarity or owe their existence outright to their trigonometric interpretation. Do write the sibtraction formula for sine in "geometric terms".

So I believe that that proof is made significantly more transparent through the use of trigonometric functions.

>The term "trigonometric" has of course a
>historical origin, as a strict mathematical definition of
>trig functions comes from analysis (not necessarily by
>infinite series, there is a definition by inversion of
>algebraic integrals).

Neah, can't believe that. Etimologically, "trigonometry" means measuring a triangle.