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

Constructions based on the first vertex

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

The three lines (ASa), (BSb) et (CSc) are concurrent.

References

External links

• [2] the original paper
• [3] announce of the theorem
• [4] the figure made with CaRMetal.
• [5] figures made by pupils