Eight circles theorem
The Eight circles Theorem (also known as Dao's Eight circles Theorem[1]) is a theorem related to eight circles, stated as follows[1][2]:
Let A1, A2, A3, A4, A5, A6 be six points on a circle (A). Point B1 lies on circle (B). Circle (AiAi+1Bi) intersects circle (B) at a second point Bi+1 for i=1,2,3,4,5. Then, A6, B6, B1, A1 lie on a circle. Let Ci be the center of circle (AiAi+1Bi+1Bi). Then, C1C4, C2C5, C3C6 are concurrent.
Proofs
[edit]This theorem was first discovered and published by Dao Thanh Oai in the journal Crux Mathematicorum in 2013, with further proofs appearing in subsequent years[1]. Chris Fisher's lemma can be directly applied to prove this theorem [3]. A proof of Chris Fisher's lemma was provided by Michel Bataille [4]. Other proofs, utilizing advanced mathematical concepts, were presented by Gábor Gévay and Ákos G. Horváth in[5] and [6]. A purely elementary proof was given by Nguyen Chuong Chi [1]. Another purely geometric proof was presented by Nguyen Ngoc Giang and Le Viet An [7].
Dual theorem
[edit]The dual theorem of the Eight circles theorem is stated as follows:
In the statement of the Eight circles Theorem, if we denote the circle (AiAi+1Bi+1Bi) as (Ci). If circle (C1) intersects circle (C4) at two points (D1, D4), circle (C2) intersects circle (C5) at two points (D2, D5), and circle (C1) intersects circle (C4) at two points (D3, D6), then the six points D1, D2, D3, D4, D5, D6 are concyclic[8].
Degenerations of the Eight circles theorem and its dual
[edit]
The Eight circles theorem and its dual can degenerate into Brianchon's theorem and Pascal's theorem when the conic in these theorems is a circle. Specifically:
- When circle (A) coincides with circle (B), the Eight circles theorem degenerates into Brianchon's theorem[8][9].
- When circle (B) degenerates into a point, the Eight circles theorem degenerates into Brianchon's theorem [9].
- When circle (B) degenerates into a point and moves to infinity, the dual of the Eight circles theorem becomes Pascal's theorem[9].
- When applying the dual of the Eight circles theorem, with circle (A) being the circumcircle and circle (B) being the First Lemoine Circle of a triangle, the circle generated by the dual theorem is known as the Dao-symedial circle[8][10].
See also
[edit]- Dao's theorem on six circumcenters
- Miquel's theorem
- Bundle theorem
- Brianchon's theorem
- Pascal's theorem
References
[edit]- ^ a b c d Nguyen Chuong Chi, A Purely Synthetic Proof of the Dao’s Eight Circles Theorem, International Journal of Computer Discovered Mathematics (IJCDM), Volume 6, 2021, pp. 87–91, ISSN 2367-7775
- ^ Dao Thanh Oai, Problem 3845, Crux Mathematicorum, 39, Issue May 2013
- ^ J. Chris Fisher, Problem 3945, Crux Mathematicorum, Volume 40, Issue May, 2014
- ^ Michel Bataille, Solution to Problem 3945, Crux Mathematicorum, Volume 41, Issue May, 2015
- ^ Gábor Gévay, A remarkable theorem on eight circles, Forum Geometricorum, Volume 18 (2018), 401--408
- ^ Ákos G.Horváth, A note on the centers of a closed chain of circles
- ^ Nguyen, Ngoc Giang; Le, Viet An (2020). Sáng tạo mới trong hình học [Novelties anh creation in geometry] (in Nhà xuất Bản đại học Quốc gia Hà Nội). Định lý 6, Chương II. Sáng tạo định lí, cách giải và kết quả hình học mới. p. 364.
{{cite book}}: CS1 maint: location (link) CS1 maint: location missing publisher (link) CS1 maint: unrecognized language (link) - ^ a b c Dao Thanh Oai, Cherng-Tiao Perng, On The Eight Circles Theorem and Its Dual, International Journal of Geometry, Vol. 8 (2019), no. 2, page 49-53
- ^ a b c Dao Thanh Oai, The Nine Circles Problem and the Sixteen Points Circle, International Journal of Computer Discovered Mathematics ISSN 2367-7775, June 2016, Volume 1, No.2, pp. 21-24.
- ^ X(5092) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(3098)Encyclopedia of Triangle Centers