Lester's Theorem

X3 = center of the circumference, X5 = center of the Feuerbach circle, X13 = first Fermat point, X14 = second Fermat point.

The set of Lester , named after June Lester , is a statement of plane Euclidean geometry , which in any non- isosceles triangle , the two Fermat points , the center of Feuerbach circle and the circumcenter are konzyklisch, so on a circle lie.

The center of this circle has the Kimberling number X (1116) and the barycentric coordinates :

{\ displaystyle (b ^ {2} -c ^ {2}) (2 (a ^ {2} -b ^ {2}) (c ^ {2} -a ^ {2}) + 3R ^ {2} (2a ^ {2} -b ^ {2} -c ^ {2}) - a ^ {2} (a ^ {2} + b ^ {2} + c ^ {2}) + a ^ {4} + b ^ {4} + c ^ {4}) \,}

{\ displaystyle (c ^ {2} -a ^ {2}) (2 (b ^ {2} -c ^ {2}) (a ^ {2} -b ^ {2}) + 3R ^ {2} (2b ^ {2} -a ^ {2} -c ^ {2}) - b ^ {2} (a ^ {2} + b ^ {2} + c ^ {2}) + a ^ {4} + b ^ {4} + c ^ {4}) \,}

{\ displaystyle (a ^ {2} -b ^ {2}) (2 (c ^ {2} -a ^ {2}) (b ^ {2} -c ^ {2}) + 3R ^ {2} (2c ^ {2} -b ^ {2} -a ^ {2}) - c ^ {2} (a ^ {2} + b ^ {2} + c ^ {2}) + a ^ {4} + b ^ {4} + c ^ {4}) \,}

literature

Clark Kimberling, "Lester Circle", Mathematics Teacher , volume 89, number 26, 1996.
June A. Lester, "Triangles III: Complex triangle functions", Aequationes Mathematicae , volume 53, pages 4–35, 1997.
Michael Trott, "Applying Groebner Basis to Three Problems in Geometry", Mathematica in Education and Research , volume 6, pages 15-28, 1997.
Ron Shail, "A proof of Lester's Theorem", Mathematical Gazette , volume 85, pages 225–232, 2001.
John Rigby, "A simple proof of Lester's theorem", Mathematical Gazette , volume 87, pages 444–452, 2003.
JA Scott, "On the Lester circle and the Archimedean triangle", Mathematical Gazette , volume 89, pages 498-500, 2005.
Michael Duff, "A short projective proof of Lester's theorem", Mathematical Gazette , volume 89, pages 505–506, 2005.
Stan Dolan, "Man versus Computer", Mathematical Gazette , volume 91, pages 469-480, 2007.