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September 10th, 2009, 05:32 AM  #1 
Newbie Joined: Sep 2009 Posts: 5 Thanks: 0  Compass and ruler construction problem
Construct a point in the line a(line through A, B) so that it's equidistant to a given point E and given line b(line through C, D). The figure of the problem is uploaded. I've thought about this problem for several nights and tried several ideas. I've been a bit frustrated. I'm not here for answers, I just wonder "Can it be solved by just compass and ruler construction?". Please let me know if it can be solved or not. If "yes", please don't let me know the construction steps, one or two hints are welcome, I'll learn to solve it by myself. If "no", I can save time for other problems. Thank you. 
September 10th, 2009, 11:02 AM  #2 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  Re: Compass and ruler construction problem
Yes, it is possible. I have not actually figured out how to do it. Are you familiar with constructible numbers? It's not too hard to prove that the distance along the line b at which to place the point (at the intersection of the perpendicular to b and line a) is constructible, so the point can be found using ruler and compass. That gives no guarantees it won't take hundreds of steps.

September 12th, 2009, 03:48 AM  #3 
Newbie Joined: Sep 2009 Posts: 5 Thanks: 0  Re: Compass and ruler construction problem
I'm not familiar with constructible number. Someone else tells me that it can be solved, in just 9 steps. But he didn't say too much. Is it really like that? Is it really constructible in 9 steps? Any hints? 
September 12th, 2009, 07:02 AM  #4 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  Re: Compass and ruler construction problem
Nine steps does not seem unreasonable. Here's how I would go about figuring them out: Let your line 'b' lie along the x axis. Put the intersection of lines 'a' and 'b' at the origin (0, 0). Call the x coordinate of point E '1', and the y coordinate 's' so that E is at (1, s). Then call the height of line 'a' at x = s the number 'r', so that the line has equation y = rx. Then you can find expressions for the point 'a' to be constructed: both the distance from E and the distance from the line will have forms in terms of r, s. Construct that length along line b and go vertically to line a. 
September 13th, 2009, 10:17 PM  #5 
Newbie Joined: Sep 2009 Posts: 5 Thanks: 0  Re: Compass and ruler construction problem
I don't understand what's meaning of "Then call the height of line 'a' at x = s the number 'r', so that the line has equation y = rx." Would you mind explain it with a figure? 
September 13th, 2009, 10:46 PM  #6 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  Re: Compass and ruler construction problem
No more typos.

September 14th, 2009, 03:38 AM  #7 
Newbie Joined: Sep 2009 Posts: 5 Thanks: 0  Re: Compass and ruler construction problem
I may understand your ideas now, but still don't understand "why should it be". I've briefly read "constructible number" in wikipedia, I think I may need more knowledge before I can really understand the "truth" behind. Thank you very much. btw, there's typo errors in your post. 
September 14th, 2009, 03:49 AM  #8 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  Re: Compass and ruler construction problem
No more typos.


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