Beyond Conway’s Concyclicity Theorem : generalization and alternatives

HAL Id: hal-02915464

https://hal.archives-ouvertes.fr/hal-02915464v2

Submitted on 31 Aug 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Beyond Conway’s Concyclicity Theorem : generalization and alternatives

David Pouvreau

To cite this version:

David Pouvreau. Beyond Conway’s Concyclicity Theorem : generalization and alternatives. 2020.

�hal-02915464v2�

Beyond Conway’s Concyclicity Theorem David Pouvreau, Quadrature , August 12th 2020

1

Beyond Conway’s Concyclicity Theorem : generalization and alternatives

Synopsis – Quadrature , 2020

David Pouvreau

Under publication in the french journal Quadrature (https://www.quadrature.info/), originally entitled « Par-delà le théorème de cocyclicité de Conway – Généralisation et alternatives ».

Submitted on July 9th 2020, accepted on August 12th 2020. Publication in the n°119 (January 2021) It is here only the synopsis of the original paper, which states the main results and provides three graphic illustrations.

1. Introduction

Let 𝐴, 𝐵 et 𝐶 be three separate and non aligned points. We label : 𝑎 = 𝐵𝐶, 𝑏 = 𝐴𝐶, 𝑐 = 𝐴𝐵 and 𝑝 the half perimeter

2

(𝑎 + 𝑏 + 𝑐) of the triangle 𝐴𝐵𝐶 𝛺 the centre of the incircle of 𝐴𝐵𝐶 and 𝑟 the radius of the incircle

(𝛼; 𝛽; 𝛾) any triplet of real numbers 𝐴′ the point of (𝐴𝐵) defined by 𝐴𝐴′ ⃗⃗⃗⃗⃗⃗ = −𝛼

𝑐

𝐴𝐵 ⃗⃗⃗⃗⃗ and 𝐴′′ the point of (𝐴𝐶) defined by 𝐴𝐴′′ ⃗⃗⃗⃗⃗⃗⃗⃗ = −𝛼

𝑏

𝐴𝐶 ⃗⃗⃗⃗⃗

𝐵′ the point of (𝐵𝐶) defined by 𝐵𝐵′ ⃗⃗⃗⃗⃗⃗⃗ = −𝛽

𝑎

𝐵𝐶 ⃗⃗⃗⃗⃗ and 𝐵′′ the point of (𝐵𝐴) defined by 𝐵𝐵′′ ⃗⃗⃗⃗⃗⃗⃗⃗ = −𝛽

𝑐

𝐵𝐴 ⃗⃗⃗⃗⃗

𝐶′ the point of (𝐶𝐴) defined by 𝐶𝐶′ ⃗⃗⃗⃗⃗⃗ = −𝛾

𝑏

𝐶𝐴 ⃗⃗⃗⃗⃗ and 𝐶′′ the point of (𝐶𝐵) defined by 𝐶𝐶′′ ⃗⃗⃗⃗⃗⃗⃗ = −𝛾

𝑎

𝐶𝐵 ⃗⃗⃗⃗⃗

2. The case (𝜶; 𝜷; 𝜸) = (𝟏; 𝟏; 𝟏) : Conway’s theorem and associated properties 2.1. Conway’s theorem

Theorem 1 (Conway)

If (𝛼; 𝛽; 𝛾) = (1; 1; 1), then the points 𝐴

, 𝐴

, 𝐵

, 𝐵

, 𝐶′ and 𝐶′′ are concyclic ; they belong to a circle (𝛤), the center of which is the point 𝛺.

2.2 Proofs of Conway’s theorem 2.2.1. A purely metric proof 2.2.2. An angular proof

2.3. Other properties associated to Conway’s configuration 2.3.1. Radius of the Conway circle

2.3.2. Properties of Conway’s hexagon

3. Generalization of Conway’s theorem

Theorem 2

Let 𝑈, 𝑉 et 𝑊 be the points of contact to the sides [ 𝐴𝐵 ] , [𝐵𝐶] et [𝐶𝐴] of the incircle of 𝐴𝐵𝐶.

Let 𝒯 be the set {( 𝛼 ; 1 + ( 𝛼 − 1 ) 𝑎/𝑏 ; 1 + ( 𝛼 − 1 ) 𝑎/𝑐 ) ∣ 𝛼 ∈ ℝ } .

1

PhD, professor of mathematics (« agrégé »), Le Tampon, La Réunion.

Beyond Conway’s Concyclicity Theorem David Pouvreau, Quadrature , August 12th 2020

2

𝐴

, 𝐴

, 𝐵

, 𝐵

, 𝐶′ et 𝐶

are concyclic and belong to a circle ( 𝛤 ) the center of which is 𝛺 if, and only if ( i ) 𝐴𝐵𝐶 is scalene and ( 𝛼; 𝛽; 𝛾 ) ∈ 𝒯 or

(ii) 𝐴𝐵𝐶 is isocele and (𝛼; 𝛽; 𝛾) ∈ 𝒯 ∪ {(0; 0; − 𝑎

𝑐 )} if it is isocele in 𝐶 (𝛼; 𝛽; 𝛾) ∈ 𝒯 ∪ {(0; − 𝑐

𝑏 ; 0)} if it is isocele in 𝐵 (𝛼; 𝛽; 𝛾) ∈ 𝒯 ∪ {(− 𝑏

𝑎 ; 0; 0)} if it is isocele in 𝐴

Moreover, when this concyclicity is realized, 𝑈, 𝑊, 𝐴′ et 𝐴′′ are concyclic, 𝑈, 𝑉, 𝐵′ et 𝐵′′ are concyclic and 𝑉, 𝑊, 𝐶′ et 𝐶′′ are concyclic.

The radius of ( 𝛤 ) is anyway 𝑅 = √(𝑝 + (𝛼 − 1)𝑎)

+ 𝑟

.

In particular, ( 𝛤 ) is the incircle of 𝐴𝐵𝐶 if, and only if 𝛼 = 1 − 𝑝/𝑎.

A case with 𝛼 = −1/2

4. The case (𝜶; 𝜷; 𝜸) = (−𝟏; −𝟏; −𝟏) : Dussau’s theorem and complements 4.1. The « anti-Conway » configuration and Dussau’s theorem

Let 𝑋, 𝑌 et 𝑍 be the points of contact to the sides [𝐴𝐵], [𝐵𝐶] and [𝐶𝐴] of the excircles of the triangle 𝐴𝐵𝐶. The lines (𝐴𝑌), (𝐵𝑍) and (𝐶𝑋) are concurrent in the Nagel point 𝐼 of 𝐴𝐵𝐶.

Theorem 3 (Dussau, 2020)

If 𝐴𝐵𝐶 is scalene and if (𝛼; 𝛽; 𝛾) = (−1; −1; −1), then the lines (𝐴

𝐶′′), (𝐵

𝐴

) and (𝐶

𝐵

) are concurrent in 𝐼.

4.2. Proof of Dussau’s theorem

4.3. Complement to Dussau’s theorem: another cocyclicity theorem Theorem 4

If 𝐴𝐵𝐶 is scalene and if (𝛼; 𝛽; 𝛾) = (−1; −1; −1), then the points 𝐴

, 𝐴

, 𝐵′ and 𝐶′′ are concyclic, the points 𝐵

, 𝐵

, 𝐶′ and 𝐴′′ are concyclic and the points 𝐶

, 𝐶

, 𝐴′ and 𝐵′′ are concyclic.

Moreover, the Nagel point 𝐼 of 𝐴𝐵𝐶 has the same power in relation to the circumscribed circles to

the three corresponding quadrilaterals, and this common power is 8𝑟

.

Beyond Conway’s Concyclicity Theorem David Pouvreau, Quadrature , August 12th 2020

3 5. Extension of Dussau’s theorem

Let’s now suppose that 𝛼, 𝛽 and 𝛾 are other than zero. If a triplet (𝛼; 𝛽; 𝛾) ≠ (−1; −1; −1) is such that (𝐴

𝐶′′), (𝐵

𝐴

) and (𝐶

𝐵

) are concurrent in 𝐼, it is called a « congruence » of (−1; −1; −1).

Theorem 5

If 𝐴𝐵𝐶 is scalene, then :

(i) There is no congruence of (−1; −1; −1) if 𝑝 ∈ {2𝑎 ; 2𝑏 ; 2𝑐 ; √2𝑏𝑐 ; √2𝑐𝑎 ; √2𝑎𝑏}.

(ii) In any other case, there exist a unique congruence of (−1; −1; −1), which is : (𝛼; 𝛽; 𝛾) = ( 𝑝

− 2𝑏𝑐

𝑝(𝑝 − 2𝑎) ; 𝑝

− 2𝑐𝑎

𝑝(𝑝 − 2𝑏) ; 𝑝

− 2𝑎𝑏 𝑝(𝑝 − 2𝑐) ) None of the three numbers of this triplet being equal to −1.

6. Conclusion