Must one of my favorite geometrical theorems come to be regarded as "trivial" or "obvious", when it is shown to have a really short and easy proof? Let C denote any closed unbounded subset of the Euclidean plane, whose interior is non-empty. One can visualize the immense variety of point sets that qualify as examples of C, and they all have positive or infinite 2-dimensional Lebesgue measure. But we have the theorem: "If C is convex, then C has infinite 2-dimensional Lebesgue measure".-Proof: Let p be an interior point of C. Then there exists a subset of C which is a closed disk D, centered at p, whose diameter is positive. Let q be a point of C not belonging to D. Consider the isoceles triangle T whose base is a diameter of D and whose opposite vertex is q. The staight line segment pq is a median and an altitude of T. T is a subset of C and the unboundedness of C implies that q can be so chosen as to make the area of T arbitrarily large-Q.E.D. Note that both the theorem and its short proof generalize readily to higher dimensional Euclidean spaces. I first realized that there was such a theorem after managing to come up with a long and unwieldy proof of it.
closed as not a real question by Todd Trimble♦, Noah S, Anthony Quas, Lee Mosher, Felipe Voloch May 26 '13 at 16:16
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The words "obvious" and "trivial" relate more to the prevailing mathematical culture than to the statement itself. Yes, the theorem you quoted is obvious in the current culture, and so is Euclid's proof, but not because the proofs are short and simple, rather because in both cases this particular way of argument became totally standard and repeats in so many places that it is essentially the first thing that comes to your mind when you look at the problem. Of course, I'm talking here of "an average mathematician". Each particular individual can manage to stay away from some aspects of general mathematical culture for a long while and then completely routine things can come as a big surprise to him (I experienced a few such surprises myself and, I believe, this experience is shared by many people). There is nothing wrong with stopping on the road and admiring the wheels of the passing cars for a while. As a matter of fact, this is quite a good exercise for your brain, IMHO (admiring=thinking of how it works and how else you could make it, not just staring). Still, most people won't share your excitement even when you show them some of the latest Lockheed Martin products instead of the car wheel: everybody got so used to the idea of the flying machines that no one is particularly impressed by anything. The ability to perceive things you see every day as unusual (and most of them are highly unusual) has to be maintained consciously by routine exercising more often than not. The same is true for the world of mathematics.