DH: Consider the relationship between circles and the equality of straight lines. A circle is defined by a collection of equal straight line segments with one common end-point.
JG: That describes the surface (area) of a circle more, rather than it does the circumference. A circle as defined by Euclid, has ONLY two components: a centre and a path (line). A circle does not consist of *anything* else. It's distinguishing feature is that every point on the circumference is located at the same distance from its centre.
DH: Two straight line segments are equal if they are radii of a common circle, or if they are the first and last of a finite sequence of straight line segments where each adjacent pair are radii of a common circle.
JG: Two straight lines are equal if the shortest distance between their endpoints is the same. Does not need to be related to a circle or any other geometric object besides the point. Newsflash: The line is defined long before the circle, triangle or any other geometric figure.
DH: That is all that Euclid's axioms and postulates tell us about the two concepts.
JG: You don't know much about Euclid at all.
DH: As it stands the two definitions form a closed loop and are absolutely useless.
JG: Which two definitions?
DH: Add in a postulate which forces certain circles to intersect and suddenly you have a powerful technique for constructing equal straight lines and the whole wonderful system becomes possible.
JG: Bollocks. You don't need intersecting circles to construct equal straight lines. You can construct equal straight lines using only one circle. Hint: Diameter.
DH: The following system models this. Take the standard model, keep the straight lines and points but define two line segments to be equal iff they are identical.
JG: They don't need redefinition. The definition of straight lines is perfect. Here you are getting muddled up with such trivial nonsense, and you think that you have what it takes to correct the greatest minds? Tsk, tsk. Don't make me laugh. From what I've read so far, you are very ignorant.
DH: Each circle has exactly one point on its circumferences, no two distinct straight line segments have the same length, but all the postulates are satisfied.
JG: That's not a circle anymore! What absolute rot!!! How are the postulates satisfied?
DH: Pasch's axiom is also obeyed, as will be many of the other implicit axioms that have been spotted over the centuries - as long as they don't involve circles or length, (e.g. order properties).
JG: Pasch's axiom:
"If a line intersects one side of a triangle internally then it intersects precisely one other side internally and the third side externally, if it does not pass through a vertex of the triangle." - Wikipedia Moronica
This rot is NOT an axiom.
"In the plane, if a line intersects one side of a triangle and misses the three vertices, then it must intersect one of the other two sides." - Mathworld
The definition of it given by Matheworld is a very obvious truth that is also *not an axiom* by any stretch of the imagination. Look, a triangle is a path that encloses an area. If a straight line enters the enclosed area, then naturally it will intersect one side and on exiting the area intersect the other side of the triangle. There is nothing else to this.
DH: Yet proposition 1 fails completely, there are no equilateral triangles.
JG: How absurd! Think they will award you with an Abel Prize for that? That's just your wrong opinion.
DH: So that answers the original question.
JG: It answers nothing, except your delusional imaginations.
DH: You can't prove, from Euclid's original axioms and postulates, that two circles which share a radius must intersect, even if you add Pasch's axiom.
JG: There is nothing AT ALL to prove. Pasch's axiom is complete rot and does not disprove ANYTHING about Euclid's axioms. Do yourself a favour and stop reading Wikipedia.