In plane geometry, the theorems of Miquel are theorems concerning of the convergent Cercle S

Statements

Theorem of the three circles : Are three circles $(C\_1)$, $(C\_2)$, $(C\_3)$ meeting in a point O , one calls M , NR and P the other points of intersection of the circles $(C\_1)$ and $(C\_2)$, $(C\_2)$ and $(C\_3)$, $(C\_3)$ and $(C\_1)$. Either has a point of $(C\_1)$ such as the line (MY) recuts $(C\_2)$ in B and the line $(Pa)$ recuts $(C\_3)$ in C . the theorem of Miquel affirms whereas the points B , NR and C are aligned.

Reciprocal : if ABC is a triangle, and if M , NR and P is three points located respectively on (AB) , (BC) and (CA) then the circles circumscribed with the triangles (AMP) , (BMN) and (CNP) meet in a point O

Theorem of the complete quadrilateral : If ABCDEF is a Complete quadrilateral then the circles circumscribed with the triangles (EAD) , (EBC) , (FAB) and (FDC) are convergent in a point O called not of Miquel

Theorem of the four circles : if $(C\_1)$, $(C\_2)$, $(C\_3)$ and $(C\_4)$ is four circles, if $A\_1$ and $B\_1$ are the intersections of $(C\_1)$ and $(C\_2)$, $A\_2$ and $B\_2$ the points of intersection of $(C\_2)$ and $(C\_3)$, $A\_3$ and $B\_3$ the intersection of $(C\_3)$ and $(C\_4)$ and $A\_4$ and $B\_4$ the intersections of $(C\_1)$ and $(C\_4)$, the points $A\_1$, $A\_2$, $A\_3$, $A\_4$ are aligned or cocyclic if and only if it is the same of the points $B\_1$, $B\_2$, $B\_3$, $B\_4$.

Theorem of the sixth circle : If ABCDE is a unspecified pentagon. If F , G , H , I , J is the points of intersection on the sides (EA) and (BC) , (AB) and (CD) , (BC) and (OF) , (CD) and (EA) , (OF) and (AB) , then the points of intersection of the five circumscribed circles with (ABF) , (BCG) , (CDH) , (DEI) , (EAJ) is located on a sixth circle which contains also the centers of the five preceding circles.

Reciprocal : if $(C\_1),$ $(C\_2)$, $(C\_3)$, $(C\_4)$, $(C\_5)$ is five circles whose centers are on a circle (C) and who then cut between neighbors on (C) the five lines uniting the points of intersection not located on (C) of a circle with its neighbors meet on the circles.

Historical remarks

Auguste Miquel published part of these theorems in the books of Liouville (Newspaper of mathematics pure and applied) in 1838. The first theorem of Miquel is a traditional result known good before him using the Théorème of the inscribed angle.

The name of not of Miquel is allotted to the point of contest of the four circles of a Complete quadrilateral but the property was known already by Jakob Steiner (1828) and even probably by William Wallace.

The theorem of the five circles (or the sixth circle) is a particular case of a general theorem stated and shown by the mathematician William Kingdon Clifford. This problem returned to the last style following a challenge launched in 2002 by the Chinese president Jiang Zemin at the time of a congress of mathematicians to Beijing in 2002. It was taken again by Alain Connes at the time of a seminar in October 2002.

External bonds

• Demonstrations of Miquel
• Demonstrations of Reciprocal Miquel
• of the theorem of the five circles
• Demonstration using the complexes

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