Hi

We define the binary operation of a projective plane as follows: Let x and y be any two points in the projective plane (think of the projective plane as un upper hemisphere). Let y' be the antipodal point of y which locates in the lower hemisphere and equal to y. Draw a line through x and y'. x*y is then defined as the end point of the line paralell to the line joining x and y' with same length and the mid point of it is the origin (Since the end points of this line are antipodal so they are equal in the projective plane).

My question is : Is it true that

x*(y*z)=(x*y)*(x*z).

If so, then please can you guide me how to prove it.

Thank you in advance