# A question concerning how mathematicians feel about theorems and their proofs. [closed]

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.

• I don't think this question is appropriate for MO, but: in defense of triviality, I'd point to the fact that part of the continuing goal of mathematics is to develop the right intuitions for its results -- so results should become easier over time. In some sense, I think the hope tends to be that theorems will become "trivial" in some sense, because that means mathematical intuition will have evolved to the point that those theorems will now seem inescapably true. – Noah Schweber May 26 '13 at 15:26
• Many if not all such elementary proofs may look obvious once discovered. But so what? One can still take pleasure in them. This question seems more suitable for a bull session over beers than a good question for MO. Voted to close. – Todd Trimble May 26 '13 at 15:27
• What IS the question?! – Amir Asghari May 26 '13 at 15:55
• The only question I saw was the first sentence, which ends in a question mark. Viz., must we regard one of Garabed's favorite geometrical theorems as trivial? – Todd Trimble May 26 '13 at 16:02