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 center of the Lester circle is also a triangle center. It is the center designated as X(1116) in the Encyclopedia of Triangle Centers. ^[9] Recently, Peter Moses discovered 21 other triangle centers lie on the Lester circle. The points are numbered from X(15535)-X(15555) in the Encyclopedia of Triangle Centers.^[10]

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. ^[11][12]

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. ^[12][13]

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. ^{[13][14][15][16]}

See also

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

References

External links

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