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

X_{13},X_{14}

, the center

X_{5}

of the nine-point circle (light blue), and the circumcenter

X_{3}

of the green triangle lie on the Lester circle (black).

References edit

^ Lester, June A. (1997), "Triangles. III. Complex triangle functions", Aequationes Mathematicae, 53 (1–2): 4–35, doi:10.1007/BF02215963, MR 1436263, S2CID 119667124
^ Kimberling, Clark (1996), "Lester circle", The Mathematics Teacher, 89 (1): 26, JSTOR 27969621
^ Shail, Ron (2001), "A proof of Lester's theorem", The Mathematical Gazette, 85 (503): 226–232, doi:10.2307/3622007, JSTOR 3622007, S2CID 125392368
^ Rigby, John (2003), "A simple proof of Lester's theorem", The Mathematical Gazette, 87 (510): 444–452, doi:10.1017/S0025557200173620, JSTOR 3621279, S2CID 125214460
^ Scott, J. A. (2003), "Two more proofs of Lester's theorem", The Mathematical Gazette, 87 (510): 553–566, doi:10.1017/S0025557200173917, JSTOR 3621308, S2CID 125997675
^ Duff, Michael (2005), "A short projective proof of Lester's theorem", The Mathematical Gazette, 89 (516): 505–506, doi:10.1017/S0025557200178581, S2CID 125894605
^ Dolan, Stan (2007), "Man versus computer", The Mathematical Gazette, 91 (522): 469–480, doi:10.1017/S0025557200182117, JSTOR 40378420, S2CID 126161757
^ Trott, Michael (1997), "Applying GroebnerBasis to three problems in geometry", Mathematica in Education and Research, 6 (1): 15–28