IR-MO 2006 Problem 8

We mean a traingle in $\mathbb Q^{n}$ , 3 points that are not collinear in $\mathbb Q^{n}$

a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$ . Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$ . b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^{n}$ it can be embedded in $\mathbb Q^{m}$ . c) Find a triangle that can be embedded in $\mathbb Q^{n}$ and no triangle similar to it can be embedded in $\mathbb Q^{3}$ . d) Find a natural $m'$ that for each traingle that can be embedded in $\mathbb Q^{n}$ then there is a triangle similar to it, that can be embedded in $\mathbb Q^{m}$ . You must prove the problem for $m=9$ and $m'=6$ to get complete mark. (Better results leads to additional mark.)