Last week while in Tel Aviv I had an interesting conversation over lunch with Leonid Polterovich and Yaron Ostrover. I happened to mention the following gem from the remarkable book A=B by Wilf-Zeilberger. The book contains the following Theorem and “proof”:
Theorem 1.4.2. For every triangle ABC, the angle bisectors intersect at one point
Proof. Verify this for the 64 triangles for which the angle at A and B are one of 10, 20, 30, , 80. Since the theorem is true in these cases it is always true.
We are asked the provocative question: is this proof acceptable? The philosophy of the W-Z method is illustrated by pointing out that this proof is acceptable if one adds for clarity the remark that the coordinates of the intersections of the pairs of angle bisectors are rational functions of degree at most 7 in the tangents of A/2 and B/2; hence if they agree at 64 points they agree everywhere.
Leonid countered with a personal anecdote. Recall that an altitude in a triangle is a line through one vertex which is perpendicular to the opposite edge. Leonid related that one day his geometry class (I forget the precise context) were given the problem of showing that the altitudes in a hyperbolic triangle (i.e. a triangle in the hyperbolic plane) meet at a single point — the orthocenter of the triangle. After the class had struggled with this for some time, the professor laconically informed them that the result obviously followed immediately from the corresponding fact for Euclidean triangles “by analytic continuation”. Philosophically speaking, this is not too far from the W-Z example, although the details are slightly more shaky — in particular, the class of Euclidean triangles are not Zariski dense in the class of triangles in constant curvature spaces, so a little more remains to be done.
Actually, one might even go back and rethink the W-Z example — how exactly are we to verify that the angular bisectors intersect at a point for the triangles in question without doing a calculation no less complicated that the general case? Let’s raise the stakes further. After some thought, we see that not only will the intersections of pairs of angle bisectors be given by rational functions of the tangents of A/2 and B/2, but the (algebraic) heights of the coefficients of these rational functions can be easily estimated, and one can therefore compute an effective lower bound on how far apart the intersections of the angle bisectors would be if they were not equal. We can then literally draw the triangles on a piece of physical paper using a protractor, and verify by eyesight that the angle bisectors appear to coincide to within the necessary accuracy. After rigorously estimating the experimental errors, we can write qed.
While I am off on a tangent, this reminds me of a discussion I once had with Michael Aschbacher about the status of arguments (in topology, say) using diagrams. This could be a computation of the fundamental group of a knot complement by Wirtinger’s algorithm, for example, or a proof that some topological 4-sphere is smoothly standard via Kirby moves. He took what I think is an extreme view, that such arguments are never mathematically valid. This is a bit of a fuzzy argument to have if one is not careful to define precisely what one means by a “diagram” — suppose (as is in fact the case) I draw a diagram by writing a (finite) .eps file in ASCII. Then a “diagram” can be taken to be a certain kind of string in a finite alphabet, and the kinds of reasoning about diagrams one is prepared to accept could also be precisely specified and formalized, and could presumably be shown to be consistent with ZFC. This shows (in some very weak sense) that it is possible to conceive of a theory of “reasoning by diagrams” which must be respectable even to Michael Aschbacher. However, in practice one “reasons using diagrams” (just as one reasons in every other context) by a combination of explicit formal rules and pre logical “leaps”: if I extend this line indefinitely, it will intersect that line here; or, if I pick up this strand and pull it behind the other strand, it will eliminate these three crossings and introduce a new crossing here. And so on. If one pursues this line of reasoning too far it starts to degenerate into questions about the reliability of short term memory, or the psychophysics of perception, which throw any kind of mathematical reasoning in question. But before reaching that point, one can argue (and Aschbacher did argue) that arguments involving diagrams are “special” because of the sheer quantity and sophistication of the pre logical leaps involved. Anyone who has seen how much effort is involved in translating e.g. the Jordan curve theorem into a formal proof system like HOL light might be prepared to concede that Aschbacher has a point.
Anyway, back to hyperbolic orthocenters. If one substitutes spherical for hyperbolic geometry, there is quite a cute proof of the existence of an orthocenter as follows. Let’s fix the unit sphere in 3-space, and let ABC be a Euclidean triangle in a plane tangent to the unit sphere and touching it exactly at the orthocenter O of ABC. Radial projection of the vertices determines a spherical triangle A’B’C’. I claim that the radial projection of the altitudes of ABC become altitudes of A’B’C’, and therefore these altitudes intersect in O, which turns out also to be the (spherical) orthocenter of the (spherical) triangle A’B’C’. To see the validity of the claim, observe first that the radial projection of a straight line in to the sphere is a great circle on the sphere; so if L is any straight line in through O, the radial projection L’ is a great circle through O. Second, note that if M is a straight line in perpendicular to L (as above), the radial projection M’ is a great circle perpendicular to L’; this follows by symmetry: reflection in the plane through the origin containing L takes M to itself and therefore M’ to itself, while fixing L’ pointwise. This proves the claim, and therefore that the (spherical) altitudes of A’B’C’ intersect at O’. By a dimension count, all spherical triangles arise in this way; qed. At this point the appeal to analytic continuation (from spherical to hyperbolic geometry) is more persuasive.