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I've drawn two random, non-parallel, straight lines on a plane. They cross over, forming two angles, $a$ and $b$, where ($a + b + a + b) = 1$ (or $360^\circ$) and $a ≤ b$. (Making $a$ either the acute angle or a right angle.)

Using only a compass and straight edge, how would I find the value of $a$? It is acceptable to use an infinite number of steps.

(This is an abstract recreational puzzle rather than a practical question. If it were practical, I'd acquire a protractor.)

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up vote 4 down vote accepted

Since you can bisect angles, erect a perpendicular to one of the lines, and use binary search to find the binary representation of how large a fraction of a right angle $a$ is.

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Ah, this method converges linearly, which makes it better than my answer which converges only logarithmically. Now is there an algorithm that converges superlinearly? – Rahul Jan 12 '12 at 13:39
You could even abandon the search when you decide you've got an answer that's "close enough" and know how much error the early exit represents. Hurrah! – billpg Jan 12 '12 at 13:53
@Rahul: That would seem to require a cost model that lets us read unboundedly many bits of information off the drawing at constant cost. I'm not sure I would consider that natural. – Henning Makholm Jan 13 '12 at 3:25

Construct the angle $qa$ by making $q$ stacked copies of $a$. Count how many full turns you make in this process, say $p$. The value of $a$ is between $2\pi \frac p q$ and $2\pi\frac{p+1}q$. Now send $q$ to infinity.

(You did say "abstract" and "[im]practical"...)

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