## Weighted Sums of Distances

Consider n points A1, ..., An and n real numbers k1, ..., kn. The geometric locus of points P, such that the combinations of the squares of distances is a constant: k1*|PA1|2 + ... + kn*|PAn|2 = k, is a circle. All the circles resulting for various k are concentric (see SquaresCombination.html for the proof).

The picture illustrates the case for n=2. D divides AB in ratio DA/DB = s/t. On the circle with center D and radius |DC''| the function f(C') = s*a2-t*b2 remains constant = k. Changing the radius of the circle c changes the value of k. The concentric circles centered at D are the level lines of the function f(C'). This is not so if we drop the squares. This is illustrated by the following picture which shows the level curves of the function g(C') = s*a-t*b.

### See Also

PedalPolygons.html
Projection_Polygon.html
SquaresCombination.html
Stewart.html

### References

Steiner, J. Werke Bd. I New York, Chelsea, 1971, pp. 15.

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