Rigby Points


Let the inner and outer Soddy triangles of a reference triangle DeltaABC be denoted DeltaPQR and DeltaP^'Q^'R^', respectively. Similarly, let the tangential triangles of DeltaPQR and DeltaP^'Q^'R^' be denoted DeltaXYZ and DeltaX^'Y^'Z^', respectively. Then the inner (respectively, outer) Rigby point Ri (respectively, Ri^') is the perspector of DeltaPQR and DeltaXYZ (respectively, DeltaP^'Q^'R^' and DeltaX^'Y^'Z^') (Oldknow 1996). The Rigby points lie on the Soddy line. They have triangle center functions


which are Kimberling centers X_(1371) and X_(1372), respectively.


Honsberger (1995) defines a different point which he calls the "Rigby point" X. Let QR be an arbitrary chord of the circumcircle of a given triangle DeltaABC, and let P be the Simson line pole of the Simson line S_P with respect to DeltaABC which is perpendicular to QR. Then it also turns out that S_Q_|_PR and S_R_|_PQ. In addition, S_A_|_BC, S_B_|_AC, and S_C_|_AB with respect to DeltaPQR.


As a result of these remarkable facts, it can be shown that the Simson lines S_P, S_Q, and S_R with respect to DeltaABC meet in the Rigby point X. Moreover, the Simson lines S_A, S_B, and S_C with respect to DeltaPQR also meet in X, and X is the orthopole of AB, BC, and AC with respect to DeltaPQR, and of PQ, QR, and PR with respect to DeltaABC. Finally, X is the midpoint of the orthocenters of DeltaABC and DeltaPQR (Honsberger 1996, p. 136).

See also

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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Kimberling, C. "Encyclopedia of Triangle Centers: X(1371)=1st Rigby point.", C. "Encyclopedia of Triangle Centers: X(1372)=2nd Rigby point.", 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.

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Rigby Points

Cite this as:

Weisstein, Eric W. "Rigby Points." From MathWorld--A Wolfram Web Resource.

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