Lester's theorem

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).

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]

Gibert's generalization

In 2000, Bernard Gibert proposed a generalization of the Lester Theorem involving the Kiepert hyperbola of a triangle. His result can be stated as follows: Every circle with a diameter that is a chord of the Kiepert hyperbola and perpendicular to the triangle's Euler line passes through the Fermat points. ^{[9] [10]}

Dao's generalizations

Dao's first generalization

In 2014, Dao Thanh Oai extended Gibert's result to every rectangular hyperbola. The generalization is as follows: Let ${\displaystyle H}$ and ${\displaystyle G}$ lie on one branch of a rectangular hyperbola, and let ${\displaystyle F_{+}}$ and ${\displaystyle F_{-}}$ be the two points on the hyperbola that are symmetrical about its center (antipodal points), where the tangents at these points are parallel to the line ${\displaystyle HG}$. Let ${\displaystyle K_{+}}$ and ${\displaystyle K_{-}}$ be two points on the hyperbola where the tangents intersect at a point ${\displaystyle E}$ on the line ${\displaystyle HG}$. If the line ${\displaystyle K_{+}K_{-}}$ intersects ${\displaystyle HG}$ at ${\displaystyle D}$, and the perpendicular bisector of ${\displaystyle DE}$ intersects the hyperbola at ${\displaystyle G_{+}}$ and ${\displaystyle G_{-}}$, then the six points ${\displaystyle F_{+}}$, ${\displaystyle F_{-},}$${\displaystyle E,}$${\displaystyle F,}$${\displaystyle G_{+}}$, and ${\displaystyle G_{-}}$ lie on a circle. When the rectangular hyperbola is the Kiepert hyperbola and ${\displaystyle F_{+}}$ and ${\displaystyle F_{-}}$ are the two Fermat points, Dao's generalization becomes Gibert's generalization. ^{[10] [11]}

Dao's second generalization

In 2015, Dao Thanh Oai proposed another generalization of the Lester circle, this time associated with the Neuberg cubic. It can be stated as follows: Let ${\displaystyle P}$ be a point on the Neuberg cubic, and let ${\displaystyle P_{A}}$ be the reflection of ${\displaystyle P}$ in the line ${\displaystyle BC}$, with ${\displaystyle P_{B}}$ and ${\displaystyle P_{C}}$ defined cyclically. The lines ${\displaystyle AP_{A}}$, ${\displaystyle BP_{B}}$, and ${\displaystyle CP_{C}}$ are known to be concurrent at a point denoted as ${\displaystyle Q(P)}$. The four points ${\displaystyle X_{13}}$, ${\displaystyle X_{14}}$, ${\displaystyle P}$, and ${\displaystyle Q(P)}$ lie on a circle. When ${\displaystyle P}$ is the point ${\displaystyle X(3)}$, it is known that ${\displaystyle Q(P)=Q(X_{3})=X_{5}}$, making Dao's generalization a restatement of the Lester Theorem. ^{[11] [12] [13] [14]}

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, S2CID   119667124
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, S2CID   125392368
4. 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
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, S2CID   125997675
6. Duff, Michael (2005), "A short projective proof of Lester's theorem", The Mathematical Gazette, 89 (516): 505–506, doi: 10.1017/S0025557200178581 , S2CID   125894605
7. Dolan, Stan (2007), "Man versus computer", The Mathematical Gazette, 91 (522): 469–480, doi:10.1017/S0025557200182117, JSTOR   40378420, S2CID   126161757
8. Trott, Michael (1997), "Applying GroebnerBasis to three problems in geometry", Mathematica in Education and Research, 6 (1): 15–28
9. Paul Yiu, The circles of Lester, Evans, Parry, and their generalizations, Forum Geometricorum, volume 10, pages 175–209, ISSN   1534-1178
10. Dao Thanh Oai, A Simple Proof of Gibert’s Generalization of the Lester Circle Theorem, Forum Geometricorum, volume 14, pages 201–202, ISSN   1534-1178
11. Ngo Quang Duong, Generalization of the Lester circle, Global Journal of Advanced Research on Classical and Modern Geometries, Vol.10, (2021), Issue 1, pages 49-61, ISSN   2284-5569
12. Dao Thanh Oai, Generalizations of some famous classical Euclidean geometry theorems, International Journal of Computer Discovered Mathematics, Vol.1, (2016), Issue 3, pages 13-20, ISSN   2367-7775
13. Kimberling, X(7668) = POLE OF X(115)X(125) WITH RESPECT TO THE NINE-POINT CIRCLE in Encyclopedia of Triangle Centers
14. César Eliud Lozada, Preamble before X(42740) in Encyclopedia of Triangle Centers

External links

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