Let $A_1, A_2, A_3, A_4$ be four distinct concyclic points with five distances $A_iA_j, i < j$, rational numbers. a) Prove that the sixth distance is also rational. b) Prove that on any circle of diameter one and for any natural number $n\geq3$ can be found n points $A_1, A_2, \ldots, A_n$, with all rational $A_iA_j$ distances. c) Prove that for any $n\geq3$ there exist in the plane $n$ noncollinear points with any distances integers. d) Prove that there are no infinite noncollinear points in the plane with the distances between any two integers. This is a problem with several sub-points all related to each other, please provide a correct and detailed answer, also please look at the attached picture for clarity, thank you in advance.

Let A1, A2, A3, A4 be four distinct concyclic points with five distances A Aj, i < j, rational numbers. a) Prove that the sixth distance is also rational. b) Prove that on any circle of diameter one and for any natural number n 3 can be found n points A, A2,..., An, with all rational AA distances. c) Prove that for any n 3 there exist in the plane n noncollinear points with any distances integers. d) Prove that there are no infinite noncollinear points in the plane with the distances between any two integers. This is a problem with several sub-points all related to each other, please provide a correct and detailed answer, thank you in advance