I Determining the number of independent equations

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Summary
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

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

berkeman

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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). :smile:
 

fresh_42

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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.
 

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