Distance and Collinearity: Proving the Relationship between Equidistant Points

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

To prove that if two points are equidistant from a third point, then the three points are collinear, we can use the concept of distance and the definition of collinearity

To prove that if two points are equidistant from a third point, then the three points are collinear, we can use the concept of distance and the definition of collinearity.

Let’s assume that we have three points A, B, and C, and points A and B are equidistant from point C. We want to prove that these three points lie on the same line.

To start, let’s consider the distance between points A and C and the distance between points B and C. Since points A and B are equidistant from point C, we can say that AC = BC.

Now, let’s assume that these three points are not collinear. If they are not collinear, then there must be another point D that lies on a different line passing through points A and C. This would mean that AD is not collinear with BC.

Since AD does not lie on the same line as BC, we can consider the distance between AD and BC. Let’s label the intersection of lines AD and BC as point E.

Since AD and BC are not collinear, we can say that AE ≠ BE.

However, since both AD and BC pass through point E, we can consider the distances AE, AC, and CE as well as the distances BE, BC, and CE.

Since AC = BC and CE is common to both these distances, we can conclude that AE = BE.

But this contradicts our previous statement that AE ≠ BE, and therefore, our assumption that the three points are not collinear is incorrect.

Hence, our assumption was incorrect, and therefore, if two points are equidistant from a third point, then the three points must be collinear.

This proves that if two points are equidistant from a third point, then the three points are collinear.

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