User:Alain Busser/Ayme's theorem
Ayme's theorem is a result about the triangle geometry dating from september 2011[1]. 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
[edit]Triangle
[edit]Let ABC (in blue) be a triangle and its circumscribed circle (in green):
Three points
[edit]Let P, Q and R be three points in the plane (not on ABC's sides):
Constructions of lines
[edit]Constructions based on the first vertex
[edit]With P
[edit]The line (AP) is the cevian of P coming from A; it cuts the opposite side in a point Pa:
With Q
[edit]In the same way, the line (AQ) cuts the opposite side in Qa:
With R
[edit]Besides, Ra is defined as the intersection of (AR) and ABC's circumscribed circle:
Circle
[edit]As the triangle PaQaRa is not flat, it has a circumscribed circle too (in red):
Point
[edit]The intersection of the two circles is made of two points; one of them is Ra.
Definition of the point related to A
[edit]The other intersection point of the two circles is denoted Sa above.
Line through A
[edit]Finally one constructs the line (ASa):
Constructions based on the second vertex
[edit]Repeating the preceding constructions with the point Q, on constructs successively
- the point Pb, intersection of (BP) and (AC);
- the point Qb, intersection of (BQ) and (AC);
- the point Rb, intersection of (BR) and the circumscribed circle;
- The circle circumbscribed to PbQbRb (in red)
- 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
[edit]Mutatis mutandis one constructs Sc related to the vertex C:
theorem
[edit]The three lines (ASa), (BSb) et (CSc) are concurrent.
References
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