Problem A. 398. (April 2006)
A. 398. Given a circle k in the plane and a line lying outside k. Moreover, a point O is given on the circle. Define the binary operator + on the points of k as follows. For arbitrary points X and Y, denote by MXY the intersection point of lines XY and . (If X=Y then take the tangent. If the lines are parallel then MXY is the ideal point.) Construct the second intersection of OMXY and k. (If the line is the tangent at O then also the second intersection is O.) Denote this point by X+Y (see the figure).
Show that operation + can be extended to the union of k, and the ideal point of such that the points with this operation form an Abelian group with the unity O; i.e. the operation has the following properties:
a) (X+Y)+Z=X+(Y+Z) for all points X, Y, Z;
b) X+Y=Y+X for all points X, Y;
c) X+O=X for all points X;
d) For an arbitrary point X there exists a point Y such that X+Y=O.
Deadline expired on 18 May 2006.