Proving the Statement: Equidistant Points from a Third Point are Collinear

If two points are equidistant from a third point, then the three points are collinear

To prove the statement that if two points are equidistant from a third point, then the three points are collinear, we need to show that the three points lie on the same straight line

To prove the statement that if two points are equidistant from a third point, then the three points are collinear, we need to show that the three points lie on the same straight line.

Let’s consider three points A, B, and C. We want to prove that if the distance between A and C is equal to the distance between B and C (i.e., AC = BC), then the three points are collinear, meaning they lie on the same line.

First, let’s assume that points A, B, and C are not collinear. This means that there exists a line segment that connects two of the points, say AB. We can now consider the midpoint M of the line segment AB.

Next, let’s examine the distances between A and M, and between B and M. Since M is the midpoint of AB, by definition, we know that AM = MB.

Now, consider the distances between A and C, and between B and C. Given that AC = BC, we have two possibilities:

1. AC > BC: In this case, we can construct a line segment from M to C. Since AC > BC, the line segment MC will be longer than MB. This contradicts our earlier statement that AM = MB.

2. AC < BC: In this case, we can construct a line segment from M to C. Since BC > AC, the line segment MC will be longer than MA. Again, this contradicts our earlier statement that AM = MB.

In both cases, we end up with a contradiction, which means our assumption that A, B, and C are not collinear must be false. Therefore, our initial statement is proved: if two points (A and B) are equidistant from a third point (C), then the three points (A, B, and C) are collinear.

Hence, if the distances between two points and a third point are equal, it implies that the three points lie on the same straight line.

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