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I know that there is a solution to this topic using algebra (for example, this post).

But I would like to know if there is a geometric proof to show this impossibility.

Thanks.

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    Not really, no, the nature of the problem turns out to be purely algebraic. In particular, you probably need to be able to show that $\pi$ is transcendental.2012-09-13

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No such proof is known. Note that this would in fact be meta-geometric: You do not give a construction of an object from givens, but you make a statement about all possible sequences of operations with your drawing instruments. Therefore it is a good idea to classify all points constructable from standard given points. This set of points has no truely geometric properties (after all, they are dense in the standard Euclidean plane, hence arbitrarily good approximations can be constructed) but nice algebraic properties (algebraic numbers with certain properties).

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    How would you show *with ruler and compass* that you cannot do something with ruler and compass? There must be some meta-argument "whatever I could do, it doesn't help". The meta-argument *about* what can be done with ruler and compass cannot be made with ruler and compass. Well, maybe by something like geometric Goödelisation, but then such a translation of an algebraic proof would not be enlightening; just as the explicit construction of the 65537-gon published by J.G.Hermes in 1894 can't be considered any source of insight beyond the algebraic proof of existence of such a construction.2012-09-14