Recall that the Ruler Axiom states that for any line m, the points on m can be put in correspondence with the real numbers, so that if points A and B correspond to numbers a and b, the the distance |AB| = the absolute value |b-a|.
Also, the triangle inequality has a corollary that points A, B, C are collinear if one of the distances |AB|. |BC|, |CA| is the sum of the other two.
Suppose point C is on line AB, distinct from A and B, and corresponds to number c. Then |AC| + |CB| = |AB| if and only if c is between a and b (i.e., either a < c < b or b < c < a). In this case, we say that C is on segment AB.
Suppose in each case that A, B, C are points. In each case there is a ruler on line AB and numbers a and b correspond to A and B. Answer these questions: (1) are the points collinear? (2) if a and b are given, what number c corresponds to C?
Suppose a = 1, b = 3, |AC| = 6, |BC| = 4.
Suppose a = -1, b = 4, |AC| = 6, |BC| = 3.
Suppose a = 970, b = -23, |AC| = 80, |BC| = 913.
Suppose a = 5, b = 11, |AC| = |BC| = (1/2)|AB|.
Suppose a = a, b = b, |AC| = |BC| = (1/2)|AB|.
Suppose a = a, b = b, |AC| = (1/3)|AB| and |BC| = (2/3)|AB|.
Answer this related question.
If M is the midpoint of AB, and a ruler on line AB corresponds
M = 3 and A to x, what number x' does B correspond to?
(Note: B is called the point reflection of A with center M.)