Determining the number of independent equations

[guideonl]

TL;DR

geometry, determining number of independet equations

Hi everyone,
In geometry problems involved with triangles (inside a circle in my case), I can write different equations based on some known measures using pythagoras theorem, similarity , trigonometric and geometric relations. But, if I write all these equations I propably get some dependent equations.
My question is: how can I avoide depending equations? in other words, for n given triangles, how many independet equations/relations i can write?
Also, please note if there are dependent equations/relations in which using one- you may not use the other, otherwise you get dependent equations.

Thanks

 

[fresh_42]

The important part first: I do not know an answer.

In geometry we have usually linear and quadratic equations, that is a set of polynomials $S=\{\,p_1,\ldots,p_n\,\}$ which generate an ideal $\mathcal{I} \subseteq \mathbb{R}[X_1,\ldots,X_m]$.
Now there are some problems: the $p_i$ are not linear, the variables are not independent, i.e. we have uniqueness up to $\mathcal{I}$ and we do not know $\mathcal{I}$, and there are many variables: angles and distances. A geometric statement is thus the question whether a certain polynomial is in $\mathcal{I}$ or not.

I guess the main problem will be $\mathcal{I}$ and your question is: How can we find a minimal system of generating polynomials for $\mathcal{I} ?$ This leads via algebraic geometry into ring theory. However, there is no general way to determine the generators of $\mathcal{I}$ that I knew of, other than in the case of linear polynomials.

 

[guideonl]

Thank you fresh_42...
Well, you took the problem far away.. I meant using basic geometry & algebra rules...

 

[berkeman]

Thank you fresh_42...
Well, you took the problem far away.. I meant using basic geometry & algebra rules...

Your thread is labeled with an "A" prefix, meaning that you want the discussion at the Advanced / graduate school level. I'll change it for you now to "I" = Intermediate (undergraduate level).

 

[fresh_42]

Thank you fresh_42...
Well, you took the problem far away.. I meant using basic geometry & algebra rules...

I ran so many times in complicated circles, arriving with an equation I could have started with, that I doubt there is an easy solution. You can only list all variables and attach to every variable what you know, e.g. given a triangle intersected by one of its heights, then we have $\alpha+\beta+\gamma =\pi$ but may only add one of the partial triangles to the list of equation. However, we might need an angle in the other part and all of a sudden we have a redundancy.

 

[fresh_42]

Thank you fresh_42...
Well, you took the problem far away.. I meant using basic geometry & algebra rules...

Here is the corresponding framework: Gröbner bases.