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Given a triangle , construct the contact triangle . Now extend lines parallel to the sides of the contact triangle from the Gergonne point. These intersect the triangle in the six points , , , , , and . C. Adams proved in 1843 that these points are concyclic in a circle now known as the Adams' circle.

Adams' circle is a central circle with circle function

which does not correspond to any notable triangle center. Its radius is the complicated expression

where is the inradius and is the semiperimeter of the reference triangle and

The center of Adams' circle is the incenter of (Honsberger 1995, pp. 62-74).

No notable triangle centers lie on Adams' circle.

Extend the segments , , and to form a triangle . Then the Gergonne point of is the symmedian point of , and Adams' circle of is the first Lemoine circle of (Honsberger 1995, p. 98).

See also

Contact Triangle, Gergonne Point

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References

Adams, C. Die Lehre von den Transversalen. Winterthur, Switzerland: Steiner'schen Buchhandlung, 1843.Honsberger, R. "A Real Gem." §7.4 (v) in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 62-64 and 98, 1995.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Mackay, J. "Symmedians of a Triangle and Their Concomitant Circles." Proc. Edinburgh Math. Soc. 14, 37-103, 1896.

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Adams' Circle

Cite this as:

Weisstein, Eric W. "Adams' Circle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AdamsCircle.html

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