Many budding geometers feel a twinge (or more) of excitement when they learn to trisect an arbitrary line segment. Not because it's necessarily an elegant construction—there are constructions for dividing a segment at any integer ratio that are perfectly formulaic—but because it seems to tiptoe towards the taboo, perhaps to create somehow the so-called impossible: The trisection of an arbitrary angle.
I certainly did. The compass and straightedge afford a lot of leverage; surely they can wrest a trisection from segment to angle. No exclamation was ever sweeter than Ha! And they said it couldn't be done!
Unfortunately, as many as have exclaimed thus over the trisection problem have done so incorrectly. A true solution remains a long way off, for algebraic reasons: As explained by m_turner, trisecting an arbitrary angle requires solving a cubic, and the compass and rule only handle quadratics and linears (respectively). But the segment-trisection approach continues to entice, because it looks so reasonable, so much of the time.
Here's why: People always try it on relatively small angles.
By some quirk of human psychology, we think of acute angles as generic. By another quirk of human psychology, we tend to pick test cases that tell us what we already 'know', providing less advantage than we deserve. And by a quirk of geometry, the smaller the angle the more it looks like we've done it. Here, a triple pitfall for the clever and eager.
Trisecting an angle is equivalent to trisecting the arc between its rays. The arc is approximated by a line segment joining the two rays at the same points. So trisecting the segment approximately trisects the arc and the angle. The smaller the angle is, the nearer the arc falls to the segment, and the better the trisection. If the angle is fairly small, the approximation is better than the accuracy of your compasses, and if the angle is zero, the trisection is exact.
But if the angle is large ... well, let's try it. Empirical counterexamples are fine, it's just empirical proofs you have to be wary of. So rule an arbitrary line, and mark off points ABCD on it at equal intervals:
-------A-----B-----C-----D------
Construct the perpendicular bisector of AD (call it EF):
E
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-------A-----B--+--C-----D------
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F
Now we can pick any point G on line EF, and AD will be the line segment that approximates the arc. If we pick a G far from the line, the angle is small, and the trisection looks OK:
E
G
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-------A-----B--+--C-----D------
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F
You'll have to visualize the line segments connecting G to A, B, C, and D. Visualize also the arc AD with center G: Pretty close to the line. But what if G is much closer?
E
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G
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-------A-----B--+--C-----D------
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F
Now it's pretty clear that angle BGC is larger than AGB and CGD are. Keep moving G closer, and BGC continues to increase, approaching 180°, whereas the other two decrease towards 0°. So this is not a trisection. Visualizing the arc shows why: It's substantially longer than the segment, and more of its length is matched to the BC segment than to segments AB or CD.
Thus concludes our foray into geometry. Tune in next week to Scientific European Frontiers, where you'll hear Kepler say,
"I've got epicycles coming out my nose, and I'm not gonna take it anymore!"