Suppose A,( A is not equal to B)B ∈ A2. Is it true that f: X → centre (A, B, X) is an affine map? Is it bijective?
As per my understanding, we have to show that F(x) = Mx + c to prove the affine map condition(like some translation is happening). But I do not understand how to prove it here. Can anyone please suggest ways to solve it?
Supposed by 'center' you mean the center of mass (intersection of medians), then $$f(x)=\frac{A+B+x}3=\frac13x+\frac13(A+B)$$ and it has an inverse $$x\mapsto 3\left(x-\frac{A+B}3\right)\,.$$