[alogo] Squares Combination

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.
In fact, the simplest case (n=2) follows from Stewart's theorem (see Stewart.html ).

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Setting B=A, C=A2, D = P, r = k1, s=k2 => (k1+k2)*PD2 = k1*|PA1|2+k2*|PA2|2 -((k1*k2)/(k1+k2))*|A1A2|2. Last term is independent from P and shows that the locus of P is a circle:
(k1+k2)*PD2 = k - ((k1*k2)/(k1+k2))*|A1A2|2, centered at D, hence independent of the value of k.
The previous equality reduces also inductively the sum for a general n. In this case replace simply the two first terms k1*|PA1|2+k2*|PA2|2 with their equal:
(k1+k2)*PD2 +((k1*k2)/(k1+k2))*|A1A2|2.
D is a point independent of the value of k and the given equation:
k1*|PA1|2 + ... + kn*|PAn|2 = k, reduces to
(k1+k2)*PD2 + k3*|PA3| + ... + kn*|PAn| = k - ((k1*k2)/(k1+k2))*|A1A2|2.
The proposition follows by repeatedly applying the previous step.

See Also



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

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