# Betweenness and Elementary Figures : Three Collinear Points

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Let A = (x1,y1), B = (x2,y2), and C = (x,y) be three collinear points in the Euclidean Plane with x1<x2. Prove that A-C-B iff x1<x<x2.

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##### Solution Summary

A proof involving collinear points is provided and the details are provided.

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Proof:

If C is between A and B, then AC+CB=AB. So (AC+CB)^2=AB^2, we have

(x-x1)^2+(y-y1)^2+(x2-x)^2+(y2-y)^2+2AC*CB=(x1-x2)^2+(y1-y2)^2

This implies ...

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