Question about the projective plane

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

Re: Question about the projective plane

Hey student2011.

For those of us who are not familiar with x*y or projective spaces (like me), can you please point out what this is?

Re: Question about the projective plane

Hi Chiro

projective space can be thought of as a sphere with anipodal points are identified. It may be represented as a hemisphere around whose rim opposite points are similarly identified.

A binary operation on a set X is a function sends an element of the cartesion product X \times X to X. We say that the set X is closed under the binary operation.

Let me ask another question which is same as the question posted above. I guess they have same answer.

Re: Question about the projective plane

Let x and y be any two points in a sphere (S^2). Let y' denote the antipodal point of y, then x,y and y' define a circle on a sphere. define a binary operation on S^2 such that x*y is the mid point of the arc joining x and y in the circle through x,y and y'.

My question is show that x*(y*z)=(x*y)*(x*z) for x,y and z belong to the sphere.

Can we consider the arc joining x and y as a line segment joining x and y and then the mid point of the arc is the mid point of that line segment? if so then I can prove it by usin triangles theories.

Please help me

Thank you in advance