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^[1]
Dao's six point circle theorem
Dao's theorem on three Euler lines

Dao's eight circles problem

File:Dao's probelm on eight circles.svg

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.^[2]

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

File:Dao's theorem on six circumcenter.svg

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.^[3]^[4] 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

File:A generalization Lester theorem1.svg

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 $H G$ ,

2-Let $K_{+}$ and $K_{-}$ two points on the hyperbola the tangents at which intersect at a point $E$ on the line $H G$ . If the line $K_{+} K_{-}$ intersects $H G$ at $D$ , and the perpendicular bisector of $D E$ intersects the hyperbola at $G_{+}$ and $G_{-}$ , then the six points $F_{+}, F_{-}, E, F, G_{+}, G_{-}$ lie on a circle ^[5]

Special case: Given triangle $A B C$ and the Kiepert hyperbola of $A B C$ , 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 $A B C$ . This theorem is Lester's theorem.

Dao's theorem on the arbelos

We can see this theorem in ^[1]^[6]^[7]^[8]^[9]

Dao six point circle

This theorem states that the centers of the six circles tangent to the medians of $△ A B C$ at the centroid and through the vertices of the triangle are concyclic. We can see three sites sourse with independent proofs of this circle.^[10]^[11]^[12]^[13] Center of this circle is X(5569) in Kimberling center ^[14]

Dao's theorem on concurrence of three Euler line

File:X(4240) in Kimberling center.svg

X(4240) in Kimberling center

We can see this theorem in.^[15]^[16]^[17] The special point of this theorem refer to the point X(4240) in Kimberling center ^[18]

External links

Online catalogue of Archimedean circles

Notes

↑ ^1.0 ^1.1 Dao Thanh, Oai (2014). "Two pairs of Archimedean circles in the arbelos". In Yiu, Paul. Forum Geometricorum (PDF). 14. pp. 201–202. ISSN 1534-1178. Search this book on
↑ Dao Thanh, Oai (2014). "Issue 5, Problem 3845". In Shawn, Godin. Crux Mathematicorum. 39. ISSN 1496-4309. Search this book on
↑ Dergiades, Nikolaos (2014). "Dao's Theorem on Six Circumcenters associated with a Cyclic Hexagon". In Yiu, Paul. Forum Geometricorum (PDF). 14. pp. 243–246. ISSN 1534-1178. Archived from the original (PDF) on 2014-10-06. Search this book on
↑ Cohl, Telv (2014). "A purely synthetic proof of Dao's theorem on six circumcenters associated with a cyclic hexagon". In Yiu, Paul. Forum Geometricorum. 14. pp. 261–264. ISSN 1534-1178. Archived from the original on 2014-10-06. Search this book on
↑ Dao Thanh, Oai (2014). "A Simple Proof of Gibert's Generalization of the Lester Circle Theorem". In Yiu, Paul. Forum Geometricorum (PDF). 14. pp. 201–202. ISSN 1534-1178. Archived from the original (PDF) on 2015-10-10. Search this book on
↑ Bogomolny, Alexander, ed. (2014). Dao's Archimedean Twins. Search this book on
↑ Bogomolny, Alexander, ed. (2014). Dao's Archimedean Twins - Second Pair. Search this book on
↑ Bogomolny, Alexander, ed. (2014). Dao's Archimedean Twins - Third Pair. Search this book on
↑ van Lamoen, Floor, ed. (2014). Circles (A61a) and (A61b): Dao pair. Search this book on
↑ Vlamst (2014). Bogomolny, Alexander, ed. Dao's Six Point Circle. Search this book on
↑ Tran Quoc Nhat Han (2013). Dien dan toan hoc, Viet nam, ed. Duong tron Dao Thanh Oai. Search this book on
↑ Dao Thanh, Oai (2014). "A synthetic proof of A.Myakishev's generalization of van Lamoen circle and an apllication". In Barbu, Catalin. International Journal of Geometry (PDF). 3. pp. 74–80. ISSN 2247-9880. Search this book on
↑ Castellsaguer, Quim, ed. (2014). r2042. Search this book on
↑ Moses, Peter (2013). Clark, Kimberling, ed. X(5569)=Center of the Dao 6 point circle. Search this book on
↑ Cohl, Telv (2014). "Dao's theorem on concurrence of three Euler lines". In Barbu, Catalin. International Journal of Geometry (PDF). 3. pp. 70–73. ISSN 2247-9880. Search this book on
↑ Montesdeoca, Angel, ed. (2014). Inversos de los puntos de corte de una recta y un triángulo. Search this book on
↑ Montesdeoca, Angel, ed. (2014). La isocúbica no-pivotal nK(X577,X2,X3). Search this book on
↑ Kimberling, C. (2014). X(4240)=Dao twelve Euler lines point. Search this book on

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.

[dao.arbel2014-1] 1.0 ^1.1 Dao Thanh, Oai (2014). "Two pairs of Archimedean circles in the arbelos". In Yiu, Paul. Forum Geometricorum (PDF). 14. pp. 201–202. ISSN 1534-1178. Search this book on

[2] Dao Thanh, Oai (2014). "Issue 5, Problem 3845". In Shawn, Godin. Crux Mathematicorum. 39. ISSN 1496-4309. Search this book on

[3] Dergiades, Nikolaos (2014). "Dao's Theorem on Six Circumcenters associated with a Cyclic Hexagon". In Yiu, Paul. Forum Geometricorum (PDF). 14. pp. 243–246. ISSN 1534-1178. Archived from the original (PDF) on 2014-10-06. Search this book on

[4] Cohl, Telv (2014). "A purely synthetic proof of Dao's theorem on six circumcenters associated with a cyclic hexagon". In Yiu, Paul. Forum Geometricorum. 14. pp. 261–264. ISSN 1534-1178. Archived from the original on 2014-10-06. Search this book on

[5] Dao Thanh, Oai (2014). "A Simple Proof of Gibert's Generalization of the Lester Circle Theorem". In Yiu, Paul. Forum Geometricorum (PDF). 14. pp. 201–202. ISSN 1534-1178. Archived from the original (PDF) on 2015-10-10. Search this book on

[6] Bogomolny, Alexander, ed. (2014). Dao's Archimedean Twins. Search this book on

[7] Bogomolny, Alexander, ed. (2014). Dao's Archimedean Twins - Second Pair. Search this book on

[8] Bogomolny, Alexander, ed. (2014). Dao's Archimedean Twins - Third Pair. Search this book on

[9] van Lamoen, Floor, ed. (2014). Circles (A61a) and (A61b): Dao pair. Search this book on

[10] Vlamst (2014). Bogomolny, Alexander, ed. Dao's Six Point Circle. Search this book on

[11] Tran Quoc Nhat Han (2013). Dien dan toan hoc, Viet nam, ed. Duong tron Dao Thanh Oai. Search this book on

[12] Dao Thanh, Oai (2014). "A synthetic proof of A.Myakishev's generalization of van Lamoen circle and an apllication". In Barbu, Catalin. International Journal of Geometry (PDF). 3. pp. 74–80. ISSN 2247-9880. Search this book on

[13] Castellsaguer, Quim, ed. (2014). r2042. Search this book on

[14] Moses, Peter (2013). Clark, Kimberling, ed. X(5569)=Center of the Dao 6 point circle. Search this book on

[15] Cohl, Telv (2014). "Dao's theorem on concurrence of three Euler lines". In Barbu, Catalin. International Journal of Geometry (PDF). 3. pp. 70–73. ISSN 2247-9880. Search this book on

[16] Montesdeoca, Angel, ed. (2014). Inversos de los puntos de corte de una recta y un triángulo. Search this book on

[17] Montesdeoca, Angel, ed. (2014). La isocúbica no-pivotal nK(X577,X2,X3). Search this book on

[18] Kimberling, C. (2014). X(4240)=Dao twelve Euler lines point. Search this book on

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Dao's theorem

Contents

Dao's eight circles problem

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

Dao's theorem on a rectangular hyperbola

Dao's theorem on the arbelos

Dao six point circle

Dao's theorem on concurrence of three Euler line

See also

External links

Notes

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