Dao's theorem

In Euclidean plane geometry, Dao's theorem may refer to any of several theorems associated with mathematician Dao Thanh Oai:

• Dao's eight circles problem, a generalization of Brianchon's theorem, proved by Luis González.
• Dao's generalization to Goormaghthigh's generalization of the Droz-Farny line theorem
• Dao's six circumcenter theorem, a generalization of Kosnita's theorem
• Dao's theorem on a rectangular hyperbola, a generalization of Lester's theorem
• Dao's theorem on the arbelos
• Dao's six point circle theorem
• Dao's theorem on three Euler lines

Dao's eight circles problem

Dao's problem on eight circles

This problem states that Let $A_{1},A_{2},A_{3},A_{4},A_{5},A_{6}$ lie on a circle and $B_{1},B_{2},B_{3},B_{4},B_{5},B_{6}$ lie on another circle. If the quadruples $A_{i},A_{i+1},B_{i},B_{i+1}$ lie on a circle with centres $C_{i}$ for $i=1,..,5$, then $A_{6},A_{1},B_{6},B_{1}$ lie on a circle $(C_{6})$ and furthermore $C_{1}C_{4},C_{2}C_{5},C_{3}C_{6}$ are concurrent.

• Special case: When six points $B_{1},B_{2},B_{3},B_{4},B_{5},B_{6}$ at center of circle $(A_{1}A_{2}A_{3}A_{4}A_{5}A_{6})$ this theorem is Brianchon theorem

Dao’s theorem on six circumcenters associated with a cyclic hexagon

Dao’s theorem on six circumcenters associated with a Cyclic Hexagon

This theorem appear in cut the knot since 12.2013 but has no solution. In 10.2014, Nikolaos Dergiades given elegant proof and Telv Cohl given synthetic proof for this theorem. This theorem states that:

Let $A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}$ be a cyclic hexagon. Let $B_{1}=A_{1}A_{6}\cap A_{2}A_{3}$, $B_{2}=A_{1}A_{2}\cap A_{3}A_{4}$, $B_{3}=A_{2}A_{3}\cap A_{4}A_{5}$, $B_{4}=A_{3}A_{4}\cap A_{5}A_{6}$, $B_{5}=A_{4}A_{5}\cap A_{1}A_{6}$, $B_{6}=A_{5}A_{6}\cap A_{1}A_{2}$. Let $C_{1},C_{2},C_{3},C_{4},C_{5},C_{6}$ be the circumcenter of six triangles: $A_{1}A_{2}B_{1}$, $A_{2}A_{3}B_{2}$, $A_{3}A_{4}B_{3}$, $A_{4}A_{5}B_{4}$, $A_{5}A_{6}B_{5}$ $A_{6}A_{1}B_{6}$ respectively. Then $C_{1}C_{4},C_{2}C_{5},C_{3}C_{6}$ are concurrent

• Special case: When the hexagon $A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}$ collapses to a triangle, this theorem is Kosnita theorem.

Dao's theorem on a rectangular hyperbola

Dao's theorem on a rectangular hyperbola

This problem states that let $H$ and $G$ lie on one branch of a rectangular hyperbola, and

1-Let $F_{+}$ and $F_{-}$ antipodal points on the hyperbola the tangents at which are parallel to the line $HG$,

2-Let $K_{+}$ and $K_{-}$ two points on the hyperbola the tangents at which intersect at a point $E$ on the line $HG$. If the line $K_{+}K_{-}$ intersects $HG$ at $D$, and the perpendicular bisector of $DE$ intersects the hyperbola at $G_{+}$ and $G_{-}$, then the six points $F_{+},F_{-},E,F,G_{+},G_{-}$ lie on a circle

• Special case: Given triangle $ABC$ and the Kiepert hyperbola of $ABC$, let $F_{+},F_{-}$ be the first and the second Fermat point, $K_{+},K_{-}$ be the inner and outer Vecten points, $H,G$ be the orthocenter and the centroid of the triangle $ABC$. This theorem is Lester's theorem.

Dao's theorem on the arbelos

We can see this theorem in

Dao six point circle

This theorem states that the centers of the six circles tangent to the medians of $\triangle ABC$ at the centroid and through the vertices of the triangle are concyclic. We can see three sites sourse with independent proofs of this circle. Center of this circle is X(5569) in Kimberling center

Dao's theorem on concurrence of three Euler line

X(4240) in Kimberling center

We can see this theorem in. The special point of this theorem refer to the point X(4240) in Kimberling center

See also

• Brianchon theorem
• Pascal theorem
• Seven circles theorem
• Kosnita theorem
• Goormaghtigh theorem
• Vecten points
• Lester's theorem
• Archimedes' quadruplets

External links

• Online catalogue of Archimedean circles

Notes

This article "Dao's theorem" is from Wikipedia. The list of its authors can be seen in its historical. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Derivation of the Cartesian form for an ellipse
• Dao's six point circle
• Abouabdillah's theorem

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