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Tripolar Coordinates

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Given a reference triangle DeltaABC and a point P, the triple (x,y,z), with x=PA, y=PB and z=PC representing the distances from P to the vertices of the reference triangle, is the tripolar coordinates of P.

The tripolar coordinates satisfy

(a^2+b^2-c^2)(x^2y^2+c^2z^2)+(a^2-b^2+c^2)(b^2y^2+x^2z^2)+(-a^2+b^2+c^2)(a^2x^2+y^2z^2)-(a^2x^4+b^2y^4+c^2z^4)-a^2b^2c^2=0 (y^2+z^2-a^2)^2x^2+(x^2+z^2-b^2)^2y^2+(x^2+y^2-c^2)^2z^2-(y^2+z^2-a^2)(x^2+z^2-b^2)(x^2+y^2-c^2)-4x^2y^2z^2=0

(Euler 1786).

Given p:q:r, the number of points having tripolar coordinates satisfying x:y:z=p:q:r depends on ap, bq and cr being the sides of a triangle (two points), a degenerate triangle (one point) or not a triangle (zero points) (Bottema 1987)

The following table summarizes the tripolar coordinated for a number of named centers.

X_ncentertripolar coordinates
X_1incenter I(sqrt((bc(-a+b+c))/(a+b+c)),sqrt((ac(a-b+c))/(a+b+c)),sqrt((ab(a+b-c))/(a+b+c)))
X_2triangle centroid G(1/3sqrt(2(b^2+c^2)-a^2),1/3sqrt(2(a^2+c^2)-b^2),1/3sqrt(2(a^2+b^2)-c^2))
X_3circumcenter O(R,R,R)
X_4orthocenter H(2R|cosA|,2R|cosB|,2R|cosC|)
X_6symmedian point K((bcsqrt(2(b^2+c^2)-a^2))/(a^2+b^2+c^2),(acsqrt(2(a^2+c^2)-b^2))/(a^2+b^2+c^2),(absqrt(2(a^2+b^2)-c^2))/(a^2+b^2+c^2))

See also

Trilinear Coordinates

Portions of this entry contributed by Floor van Lamoen

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References

Bottema, O. "On the Distances of a Point to the Vertices of a Triangle." Crux Math. 10, 242-246, 1984.Bottema, O. Hoofdstukken uit de Elementaire Meetkunde, 2nd ed. Utrecht: Epsilon, pp. 33-38, 1987.Euler, L. "De symptomatibus quatuor punctorum in eodem plano sitorum." Acta Acad. Sci. Petropolitanae, 6 I, 3-18, 1786. Reprinted in Opera Omnia, Series Prima, Vol. 26, pp. 258-269.Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, 1913.Hatzipolakis, A. P. van Lamoen, F. M.; Wolk, B.; and Yiu, P. "Concurrency of Four Euler Lines." Forum Geom. 1, 59-68, 2001. http://forumgeom.fau.edu/FG2001volume1/FG200109index.html.Lalesco, T. La géometrie du triangle. Paris: Gabay, 1987.Poulain, A. "Des coordonnées tripolaires." J. des Mathématiques Spéciales, 3, 3-10, 51-55, 130-134, 155-159, and 171-172, 1889.

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Tripolar Coordinates

Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "Tripolar Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TripolarCoordinates.html

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