User:Alain Busser/Ayme's theorem

Ayme's theorem is a result about the triangle geometry dating from september 2011. It is a result about projective geometry. This theorem is due to Jean-Louis Ayme, retired mathematics teacher from Saint-Denis on Reunion island.

=Hypotheses of the theorem=

Triangle
Let ABC (in blue) be a triangle and its circumscribed circle (in green):



Three points
Let P, Q and R be three points in the plane (not on ABC's sides):



=Constructions of lines=

With P
The line (AP) is the cevian of P coming from A; it cuts the opposite side in a point Pa:



With Q
In the same way, the line (AQ) cuts the opposite side in Qa:



With R
Besides, Ra is defined as the intersection of (AR) and ABC's circumscribed circle:



Circle
As the triangle PaQaRa is not flat, it has a circumscribed circle too (in red):



Point
The intersection of the two circles is made of two points; one of them is Ra.

Definition of the point related to A
The other intersection point of the two circles is denoted Sa above.

Line through A
Finally one constructs the line (ASa):



Constructions based on the second vertex
Repeating the preceding constructions with the point Q, on constructs successively


 * 1) the point Pb, intersection of (BP) and (AC);
 * 2) the point Qb, intersection of (BQ) and (AC);
 * 3) the point Rb, intersection of (BR) and the circumscribed circle;
 * 4) The circle circumbscribed to PbQbRb (in red)
 * 5) The intersection of this circle with ABCs circumscribed circle: The point Sb:



The last constructed point (Sb) is then joined to its related vertex B by a line:



Constructions based on the third vertex
Mutatis mutandis one constructs Sc related to the vertex C:



=theorem=