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Please note that the numbering of the postulates (P) is based on my geometry book. Also, I'm just in 9th grade geometry, so please don't use differential equations or some other math than basic Euclidean geomtery. This will help me better understand the concept.

This is what I did so far:

This is where I kind of get lost....Let points A and B be the end points of a line segment, AB.

By P2-3 (For any two points on a line, and a given unit of measure, there is a unique positive number called the measure of the distance between the two points) there is a defined distance between the points A and B on line segment AB

By the "definition of between", a point Q is between points A and B if and only if each of the following conditions hold.

1.) A, Q, B are collinear.

2.) AQ + BQ = AB

Condition 1: A,Q,B are collinear

By P1-1 ( Through any two points there is exactly one line) and P1-3 (There are at least two points on a line), points A, Q and B are on the same line. Therefore, by the definition of collinear (which states that points are collinear if and only if they are on the same line), A,Q and B are collinear.

Condition 2: AQ + BQ = AB, Let Q be the midpoint

Since Q is the only common point of segments AQ and BQ, then AQ and BQ both intersect at Q. Therefore, the end points are A and B. Since A, Q and B are all collinear, they form a line segment AB. Thus Q is in between points A and B in AB.

Does that prove Q is the midpoint? I don't think so because:AQ + BQ = AB

Let A=2, B=8, Q=5

l 2-5 l + l 8-5 l = l 2-8 l

6=6

By the "definition of midpoint", a point Q is the midpoint of a segment AB if and only if Q is between A and B and AQ=BQ.

1.) I haven't shown that the arithmetic I did in an attempt to show the distances between AQ and BQ are equal to that of AB works for all cases.

2.)I'm not sure I have adequately proven that A, Q and B are on the same line.