Let there be given two lines $D_1$ and $D_2$ which intersect at point $O$, and a point $M$ not on any of these lines. Consider two variable points $A\in D_1$ and $b\in D_2$ such that $M$ belongs to the segment $AB$. (a) Prove that there exists a position of $A$ and $B$ for which the area of triangle $OAB$ is minimal. Construct such points $A$ and $B$. (b) Prove that there exists a position of $A$ and $B$ for which the area of triangle $OAB$ is minimal. Show that for such $A$ and $B$, the perimeters of $\triangle OAM$ and $\triangle OBM$ are equal, and that $\frac{AM}{\tan\frac12\angle OAM}=\frac{BM}{\tan\frac12\angle OBM}$. Construct such points $A$ and $B$.