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Snapshot 1: the sum of the squares of the distances to two fixed points (-1, 0) and (1, 0) equals 3

Snapshot 2: the sum of the squares of the distances to two fixed points (-1, 0) and (1, 0) equals 4

Snapshot 3: the sum of the squares of the distances to two fixed points (-1, 0) and (1, 0) equals 16

Snapshot 4: the sum of the squares of the distances to three fixed points (-1, 0), (1, 0), and (2, -1) equals 16

We often define a circle as the locus of points that are a given distance from a given point. The given distance is the radius, and the given point is the center.

Likewise, an ellipse is often defined as the locus of points such that the sum of the distances to two fixed points is constant. The two fixed points are the foci.

Suppose we tweak these definitions and ask, "What is the locus of points such that the sum of the squares of the distances to two fixed points is constant?"

Assuming the sum is large enough, the locus is a circle.

We can generalize this: if is any positive integer, a circle is the locus of points such that the sum of the squares of the distances to fixed points is constant (again, provided the sum is large enough).

Here is the proof for the case . Write the equation for the sum of the squares of the distances to the two points and :

.

Expand, then rearrange the equation, then divide through by 2 to get this:

. (1)

Notice that are the first two terms of . We can convert into a square by adding to both sides of equation (1). This is called "completing the square". When we do this for both the and terms, we get the equation for a circle:

. (2)

The center of the circle is at the point

, .

If the right-hand side of equation (2) is a number greater than 0, then the radius is the square root of that number.

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This also generalizes to higher dimensions. For example, in 3-space, a sphere is the locus of points such that the sum of the squares of the distances to fixed points is constant.