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Musselman's Theorem

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MusselmansTheorem

Consider a reference triangle DeltaABC with circumcenter O and orthocenter H, and let DeltaA^*B^*C^* be its reflection triangle. Then Musselman's theorem states that the circles AOA^*, BOB^*, and COC^* meet in a second point that is the inverse in the circumcircle of the isogonal conjugate of the nine-point center. This point is Kimberling center X_(1157) and has center function

alpha_(1157)=(a^6-3b^2a^4-3c^2a^4+3b^4a^2+3c^4a^2-b^2c^2a^2-b^6-c^6+b^2c^4+b^4c^2)sec(B-C).

See also

Reflection Triangle

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References

Grinberg, D. "On the Kosnita Point and the Reflection Triangle." Forum Geom. 3, 105-111, 2003. http://forumgeom.fau.edu/FG2003volume3/FG200311index.html.Musselman, J. R. and Goormaghtigh, R. "Advanced Problem 3928." Amer. Math. Monthly 46, 601, 1939.Musselman, J. R. and Goormaghtigh, R. "Solution to Advanced Problem 3928." Amer. Math. Monthly 48, 281-283, 1941.Nguyen, K. L. "A Synthetic Proof of Goormaghtigh's Generalization of Musselman's Theorem." Forum Geom. 5, 17-20, 2005.

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Musselman's Theorem

Cite this as:

Weisstein, Eric W. "Musselman's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MusselmansTheorem.html

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