Lester's theorem

In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle. The result is named after June Lester, who published it in 1997,[1] and the circle through these points was called the Lester circle by Clark Kimberling.[2] Lester proved the result by using the properties of complex numbers; subsequent authors have given elementary proofs[3][4][5][6], proofs using vector arithmetic,[7] and computerized proofs.[8]

The Fermat points ${\displaystyle X_{13},X_{14}}$, the center ${\displaystyle X_{5}}$ of the nine-point circle (light blue), and the circumcenter ${\displaystyle X_{3}}$ of the green triangle lie on the Lester circle (black).

See also

• Parry circle
• Shape § Similarity classes
• van Lamoen circle

References

1. Lester, June A. (1997), "Triangles. III. Complex triangle functions", Aequationes Mathematicae, 53 (1–2): 4–35, doi:10.1007/BF02215963, MR 1436263
2. Kimberling, Clark (1996), "Lester circle", The Mathematics Teacher, 89 (1): 26, JSTOR 27969621
3. Shail, Ron (2001), "A proof of Lester's theorem", The Mathematical Gazette, 85 (503): 226–232, doi:10.2307/3622007, JSTOR 3622007
4. Rigby, John (2003), "A simple proof of Lester's theorem", The Mathematical Gazette, 87 (510): 444–452, doi:10.1017/S0025557200173620, JSTOR 3621279
5. Scott, J. A. (2003), "Two more proofs of Lester's theorem", The Mathematical Gazette, 87 (510): 553–566, doi:10.1017/S0025557200173917, JSTOR 3621308
6. Duff, Michael (2005), "A short projective proof of Lester's theorem", The Mathematical Gazette, 89 (516): 505–506, doi:10.1017/S0025557200178581
7. Dolan, Stan (2007), "Man versus computer", The Mathematical Gazette, 91 (522): 469–480, doi:10.1017/S0025557200182117, JSTOR 40378420
8. Trott, Michael (1997), "Applying GroebnerBasis to three problems in geometry", Mathematica in Education and Research, 6 (1): 15–28

External links

• Weisstein, Eric W. "Lester Circle". MathWorld.