Pole (geometry)

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Definition and construction of two geometric poles, P1 and P2, and their corresponding polars, Π1 and Π2. Since P1 and P2 are inverses of each other, the product of their distances to the circle center O equals the squared radius of the circle. This may be shown using the two similar triangles illustrated below. Given the point P2, the polar may be found by drawing a circle centered on M (the midpoint of OP2) and passing through O and P2; the intersection points with the circle define the polar Π2.
Definition and construction of two geometric poles, P1 and P2, and their corresponding polars, Π1 and Π2. Since P1 and P2 are inverses of each other, the product of their distances to the circle center O equals the squared radius of the circle. This may be shown using the two similar triangles illustrated below. Given the point P2, the polar may be found by drawing a circle centered on M (the midpoint of OP2) and passing through O and P2; the intersection points with the circle define the polar Π2.

In Euclidean geometry, the pole of a line L in a circle L is a point P that is the inversion of the point Q on L that is closest to the center of the circle. Conversely, the polar of a point P in a circle C is the line L such that its closest point Q to the circle is the inversion of P.

The relationship between poles and polars is reciprocal. Thus, if a point Q is on the polar A of a point P, then the point P must lie on the polar B of the point Q. The two polar lines A and B need not be parallel.

Poles and polars were defined by Joseph Diaz Gergonne and play an important role in his solution of the problem of Apollonius.

[edit] Reference

  • Johnson RA (1960). Advanced Euclidean Geometry: An Elementary treatise on the geometry of the Triangle and the Circle. New York: Dover Publications, pp. 100–105.