Suppose the concept of distance between points in the plane is already defined. Write a careful definition for one point to be between two other points. Your definition should begin Suppose A; B; C are points in the plane. We say that C is between A and B provided. . . . Note: Since you are crafting this definition, you have a bit of flexibility. Consider the possibility that the point C might be the same as the point A or B, or even that A and B might be the same point. Personally, if A and C were the same point, I would say that C is between A and B (regardless of where B may lie), but you may choose to design your definition to exclude this possibility. Whichever way you decide is fine, but be sure your definition does what you intend. Note further: You do not need the concept of collinearity to define between. Once you have defined between, please use the notion of between to define what it means for three points to be collinear. Your definition should begin Suppose A; B; C are points in the plane. We say that they are collinear provided. . . . Note even further: Now if, say, A and B are the same point, you certainly want your definition to imply that A, B, and C are collinear.

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