XXII OBM

Line r passes through the corner $A$ of a sheet of paper and makes an angle $\alpha$ with the horizontal border, as shown in figure 1. In order to divide $\alpha$ into three equal parts we proceed as follows:

a) initially we mark two points $B$ and $C$ on the vertical border such that $AB=BC$; through $B$ we draw a line $s$ parallel to the border (figure 2);

b) after that, we fold the sheet so as to make $C$ coincide with a point $C^{\prime }$ on the line $r$ and $A$ with a point $A^{\prime }$ on line $s$ (figure 3); we call $B^{\prime }$ the point which coincides with $B$.

Figure 1 Figure 2 Figure 3

Show that lines $A{A}^{\prime }$ and $A{B}^{\prime }$ divides angle $\alpha$ into three equal parts.