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Show that the affirmation

"For every three points (not on the same line) on the plane, one can find a circle that contains these three points"

is equivalent to

"Given a line and a point out of this line, there is only one line passing through this point and parallel to the first one".

I was able to show that the second affirmation implies the first one:

Consider a triangle ABC. Let M be the midpoint of AB. Mark P such that the angle PMA = 90° = PBM. Now, let N be the midpoint of BC. Using the second statement, there is only one line passing through N parallel to MP. This line can't be orthogonal to BC, otherwise we would have a triangle with two rect angles. So, the line passing through N perpendicular to BC meets PM in a point O. Now, it's easy to construct such a circle.

But I couldn't prove the reciprocal...

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Depends on what axioms one is working with. The circle assertion is true for the geometry of the sphere. –  André Nicolas May 4 '12 at 22:36
1  
Don't you want to edit in something about the three points not all lying in a straight line? –  Gerry Myerson May 4 '12 at 23:19
    
Thank you. I missed that. –  Br09 May 4 '12 at 23:34
    
It appears that there is no short answer. See M. J. Greenberg, 4th edition, Euclidean and Non-Euclidean Geometries, Chapter 6, Exercises 10,11 on pages 273-275, and Major Exercise 7 on page 281. In essence, you deny the parallel postulate, show quite a number of properties of the hyperbolic plane, finally show that some triangles in the hyperbolic plane lack circumscribed circles (when two sides are close enough to parallel that their perpendicular bisectors do not meet). If you come up with a short proof, let me know. I'll tell Marvin. –  Will Jagy May 5 '12 at 4:55
    
Well, R. Hartshorne, Geometry: Euclid and Beyond, exercise 33.11 on pages 302-303. He calls the circle thing Bolyai's Axiom, in this case Farkas Bolyai. He gives a pretty complete recipe. So it depends on what background you have set up when you get to the question. Marvin mostly wants you to think about it. I'm in Marvin's book, mostly pages 524-526. –  Will Jagy May 5 '12 at 5:07

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