Circle center
How many moves does it take to find the center of a (given) circle with a straightedge and compass?
All terminology in this question is explained in the first note of my straightedge and compass set. More straightedge and compass constructions can be found there.
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1 solution
1) Draw circle of arbitrary radius centered on arbitrary point on given circle, intersecting in points A, B
2) Draw same radius circle centered on A, intersecting first circle at points A1, A2
3) Draw same radius circle centered on B, intersecting first circle at points B1, B2
4) Draw line through points A1, A2
5) Draw line through points B1, B2, intersecting first line at CENTER of given circle
5 moves
This depends on having "fixed" compass that will hold a given radius (Well, maybe not---see Kesa)
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Your solution is close to the best solution for this, but not quite since it requires a fixed compass. Euclid the Game has indeed confirmed my 5 primitive move solution.
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Wen Z did refer to his "first note" about the "rules", so I went by that. But your approach is basically the same thing, but improved.
"Euclid the Game"---interesting, first time I heard about this.
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@Michael Mendrin – It's quite a cool game, and the real challenge is finding the minimum number of constructions to do each. There's stuff like constructing a regular pentagon in 10 moves. I'm still stuck on using a collapsible compass and straightedge to copy an angle in 4 constructions.
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@Sharky Kesa – The requirement that the compass be "collapsible" seems like pretty severe one---I can see how that would make a lot of ordinary compass-and-straightedge problems far more difficult and cumbersome, even though anything that can be done with a non-collapsing compass can be done with a collapsible one.
I have a solution using 5 moves (straightedge and collapsible compass).
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Choose a point A on the given circle. (0 moves)
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Construct a circle such that it intersects the circle twice (reasonable measure). Call one of these intersection points B . (1 move)
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Construct a circle centred at B passing through A . Let it intersect the circle again at C . (2 moves)
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Let D and E be the intersection points between the circle centred at A and the circle centred at B .
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Construct a circle centred at C passing through B . (3 moves)
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Let F and G be the intersection points between the circle centred at B and the circle centred at C .
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Draw ray D E . (4 moves)
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Draw ray F G . (5 moves)
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Call the intersection of these two rays O . O is the centre of the circle.
This took a total of 5 moves.
Okay sorry about that, @Calvin Lin please update the solution.
I am pretty sure this is the fastest possible. If you can find a faster one then I will be happy to update the answer.
This is Sharky Kesa's solution:
Solution using 5 moves (straightedge and collapsible compass):
Choose a point A on the given circle. (0 moves)
Construct a circle such that it intersects the circle twice (reasonable measure). Call one of these intersection points B . (1 move)
Construct a circle centred at B passing through A . Let it intersect the circle again at C . (2 moves)
Let D and E be the intersection points between the circle centred at A and the circle centred at B . Construct a circle centred at C passing through B . (3 moves)
Let F and G be the intersection points between the circle centred at B and the circle centred at C . Draw ray D E . (4 moves)
Draw ray F G . (5 moves)
Call the intersection of these two rays O . O is the centre of the circle.
This took a total of 5 moves.