![]() Consider the cones determined by each couple of spheres the section is the couple of tangent lines seen above, and the tips of the cones are the three points in the problem. You have three spheres, and if you section them through their centers with a plane you get the original three circles. Now consider the following solution: add a dimension. You may want to think a little about the problem can be solved both by plane or analytic geometry, with some effort. We get three points: prove that they are collinear. Repeat the construction for each pair of circles. For each pair of circles, there are four straight lines tangent to both take the two which leave both circles on the same side they intersect at a point. ![]() Problem: three circles in the plane, no two with the same radius, pairwise disjoint. (I learned this from Martin Gardner, proper credits might be researched if necessary). unit tangent vector.ĭon't know for sure if this example qualifies, but it certainly is a hard problem which becomes trivial from the right point of view. We will construct elements $x, y$ of $\operatorname$ are the Fourier coefficients of the position vector of the curve w.r.t. Let $q$ be a prime power such that $q-1$ is divisible by $2a$, $2b$, and $2c$. Unfortunately, it doesn't seem to be available on the archive of the list, so I will just post it here verbatim: I don't know who discovered the more modern proofs, but Derek Holt posted a proof on the group-pub that is one of the most elegant things I've ever seen. Miller, whose proof looked at lots of separate cases, and had tons of long, tedious calculations in symmetric groups (I will try and find the paper and post the reference later). I think the first person to prove this was G.A. $x$ has order $a$, $y$ has order $b$, and $xy$ has order $c$. There is a theorem in finite group theory, that if $a$, $b$, and $c$ are integers all greater than $1$, there exists a finite group $G$ with elements $x$ and $y$ such that: I decided to make this a community wiki, and I think the usual "one example per answer" guideline makes sense here. I'll post an answer which gives what I would consider to be an example. In summary, I'm looking for results that everyone thought was really hard but which turned out to be almost trivial (or at least natural) when looked at in the right way. it is fine if the machinery is extremely difficult to construct). ![]() ![]() Finally, I insist that the proofs really be quick (it should be possible to explain it in a few sentences granting the machinery on which it depends) but certainly not necessarily easy (i.e. I would also prefer results which really did have difficult solutions before the quick proofs were found. ![]() I am not as interested in problems which motivated the development of complex machinery that eventually solved them, such as the Poincare conjecture in dimension five or higher (which motivated the development of surgery theory) or the Weil conjectures (which motivated the development of l-adic and other cohomology theories). I would like to hear about some examples of problems which were originally solved using arduous direct techniques, but were later found to be corollaries of more sophisticated results. Many problems which first are solved via "direct" methods (long and difficult calculations, tricky estimates, and gritty technical theorems) later turn out to follow beautifully from basic properties of simple devices, though it often takes some work to set up the new machinery. Mathematics is rife with the fruit of abstraction. ![]()
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