
#37
Sep309, 02:01 PM

P: 891





#38
Sep309, 02:07 PM

P: 707

Maybe we need the axiom that two lines intersect in at most one point. It would imply the existence of a new point lying on both lines  a newly constructed point of space. Very cool. There still is the problem of knowing that the new line determined by the circle arcs is not parallel to the original. Do you see a way out of it? Maybe the circle arcs must intersect once in each half plane  by symmetry  and so the two points being in opposite half planes must determine a line that intersects the first. 



#39
Sep309, 03:56 PM

P: 891





#40
Sep309, 06:34 PM

Emeritus
Sci Advisor
PF Gold
P: 16,101

What exactly do you mean by "get"? The midpoint between two points exists simply by Euclidean geometry  any choice of model has nothing to do with it.
Some models might give nice nongeometric ways to describe that midpoint. For example, in any of the "usual" models of Euclidean geometry where points are pairs of real numbers, the midpoint of (a,b) and (c,d) is ((a+c)/2, (b+d)/2). You need to see a model of the type I described explicitly? Okay. . A point is an ordered pair (x,y) of real numbers . The points (x,y) and (u,v) are equal iff the ordered pairs (x,y) and (u,v) are equal . A line is an unordered pair of distinct points . A point (x,y) lies on the line {(a,b), (c,d)} iff (db)x+(ac)y=adbc . The lines {(a,b),(c,d)} and {(e,f),(g,h)} are equal iff (e,f) and (g,h) lie on {(a,b),(c,d)} . The point (c,d) lies between (a,b) and (e,f) iff (ca)²+(db)²+(ec)²+(fd)²=(ea)²+(fb)² . The line segment between (a,b) and (c,d) and the line segment between (e,f) and (g,h) are congruent iff (ca)²+(db)²=(ge)²+(hf)² This gives semantics to the five elementary undefined terms of (one presentation of) Euclidean geometry: "point", "line", "lies on", "between", and "congruent". For concreteness, I'll choose a specific meaning for "line segment" too: . A line segment is an unordered pair of distinct points . Two line segments are equal iff the unordered pairs are equal . The "line segment between P and Q" means the unordered pair {P,Q} . A point Q lies on the line segment {P,R} iff Q lies between P and R, Q is P, or Q is R. 



#41
Sep309, 08:06 PM

P: 707

What I was wondering is whether one can preserve the definition of a line as two points. My gut tells me that you can. While the mid point could be located once a metric is introduced, I was trying to avoid a metric in the spirit of keeping lines as pairs of points. Without a metric the midpoint would have to obey some symmetry  and it seems that symmetry could go a long way in determining the properties of new points. 



#42
Sep409, 07:13 PM

P: 891

A straight line is defined by an unordered pair of distinct points {P,Q} The line segment between P and Q means the the portion of the straight line extending from P to Q of the unordered pair of distinct points {P,Q} Of course both models are otherwise the same, but some one realized that the semantics of the extra language was open to varying interpertations and that one could avoid the ambiguty by omitting the extra language. Now we see that Wofsy is confused by the "more concise language". As to the meaning of "get" Wofsy meant to construct the mid point using a straight edge and compass but did not realized that the line and the line segment between P and Q are in fact given under your model of a line as a pair of unordered points {P,Q} by the statement as to which points lie thereon. As you say the exact semantics are not material once each model is understood. 


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