Pattern of Two Loci

Circles tangent to a lineCircles passing through center

According to G. Pólya, the method of finding geometric objects by intersection.

1. For example, the centers of all circles tangent to a straight line s at a given point P lie on a line t that passes through P and is perpendicular to s.

2. In addition, the circle c centered at P with radius R is the locus of the centers of all circles of radius R passing through P.

Circles tangent and passing through a center

The intersection of c and t consists of two points A and B which are the centers of two circles of radius R tangent to s at P.

Many constructions with straightedge and compass are based on this method, as, for example, the construction of the center of a given circle by means of the perpendicular bisector theorem.

See also

Cartesian Pattern

This entry contributed by Margherita Barile

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Pólya, G. Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, 2 vols. in One. New York: Wiley, 1981.

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Pattern of Two Loci

Cite this as:

Barile, Margherita. "Pattern of Two Loci." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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