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Hart Circle


HartCircle

The circle H which touches the incircles I, I_A, I_B, and I_C of a circular triangle ABC and its associated triangles. It is either externally tangent to I and internally tangent to incircles of the associated triangles I_A, I_B, and I_C (as in the above figure), or vice versa. The Hart circle has several properties which are analogous to the properties on the nine-point circle of a linear triangle. There are eight Hart circles associated with a given circular triangle.

The Hart circle of any circular triangle and the Hart circles of the three associated triangles have a common tangent circle which touches the former in the opposite sense to that which it touches the latter (Lachlan 1893, p. 254). In addition, the circumcircle of any circular triangle is the Hart circle of the circular triangle formed by the circumcircles of the inverse associated triangles (Lachlan 1893, p. 254).


See also

Associated Triangles, Circle, Circular Triangle

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References

Casey, J. "On the Equations and Properties--(1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane." Proc. Roy. Irish Acad. 9, 396-423, 1864-1866.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 43, 1971.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 127-128, 1929.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 254-257, 1893.Larmor, A. "Contacts of Systems of Circles." Proc. London Math. Soc. 23, 136-157, 1891.

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Hart Circle

Cite this as:

Weisstein, Eric W. "Hart Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HartCircle.html

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