My first reply will be an existence proof. Perhaps I shall include two existence proofs just for the heck of it. There are a lot of them
My second reply will be a hint to the construction problem.
My final reply will be one possible solution that happens to be very elegant. It may take me a while to post though since it's harder to be clear with just text.
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4 comments:
I suddenly realize that I cannot post pictures to my replies. This is a hindrance.
Well, I'll post for now and work on getting illustrations in a bit.
First, let us relax the conditions of the problem a little. It is not immediately clear if a square can be so inscribed, so let us see if we can inscribe a rectangle.
You'll find that inscribing a rectangle is actually quite a simple task. Try inscribing two rectangles, one fat, short one and one tall skinny one. Start with the fat, short one and slowly start moving it up towards the tall, skinny one. Each millimeter that you move it will make it slightly taller and slightly less fat until eventually it should equal the tall, skinny one. Since it increased continuously, at some point the height must have equaled the width, thus producing a square.
Hopefully I will have a graph and some pictures to illustrate this. I will now investigate the feasibility of posting pictures in a comment.
Hmmm. No success. Well, I think I'll just post the rest of the solutions (and repost this existence proof that I've already stated to a certain extent) in a week or so. I'm pretty much going to post when Steven says it's ok since he gets so touchy about people ruining things for him. He's such a sweetie really.
boastful:
I actually got the exact existence proof you described in relatively short order (a minute or so) after seeing the picture you posted. Yay!
thoughtful:
I didn't get it when you told me over the phone. Really shows the power of drawing (or in this case seeing) pictures to aid in visualizing the problem.
modest:
I wasn't optimistic about my chances of finding a construction, so I'm reading this now. I see that you've given me another week. It's possible I might get it by then, but I doubt it. That part looks hard!
I wonder if my feeble word-oriented brain is up to the task...
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