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# Rigby Points

Let the inner and outer Soddy triangles of a reference triangle be denoted and , respectively. Similarly, let the tangential triangles of and be denoted and , respectively. Then the inner (respectively, outer) Rigby point Ri (respectively, ) is the perspector of and (respectively, and ) (Oldknow 1996). The Rigby points lie on the Soddy line. They have triangle center functions

 (1) (2)

which are Kimberling centers and , respectively.

Honsberger (1995) defines a different point which he calls the "Rigby point" . Let be an arbitrary chord of the circumcircle of a given triangle , and let be the Simson line pole of the Simson line with respect to which is perpendicular to . Then it also turns out that and . In addition, , , and with respect to .

As a result of these remarkable facts, it can be shown that the Simson lines , , and with respect to meet in the Rigby point . Moreover, the Simson lines , , and with respect to also meet in , and is the orthopole of , , and with respect to , and of , , and with respect to . Finally, is the midpoint of the orthocenters of and (Honsberger 1996, p. 136).

Contact Triangle, First Eppstein Point, Gergonne Point, Griffiths Points, Incenter, Orthopole, Second Eppstein Point, Simson Line, Soddy Centers, Soddy Triangles, Tangential Triangle

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## References

Kimberling, C. "Encyclopedia of Triangle Centers: X(1371)=1st Rigby point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1371.Kimberling, C. "Encyclopedia of Triangle Centers: X(1372)=2nd Rigby point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1372.Honsberger, R. "The Rigby Point." §11.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 132-136, 1995.Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319-329, 1996.

Rigby Points

## Cite this as:

Weisstein, Eric W. "Rigby Points." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RigbyPoints.html