Solution of Triangle

For the triangle $$ABC$$, prove that
$$r_{1}=r\cot\dfrac{B}{2}\cot\dfrac{C}{2}$$
$$ABC$$ is a triangle.$$O$$ is a point inside the triangle so that its distance from $$A,B,C$$ is respectively $$a,b,c$$. $$L, M, N$$ are the feet of perpendiculars from $$O$$ to $$AB,BC,CA$$ respectively. $$x,y,z$$ are respectively the distances of $$O$$ from $$L,M,N$$ 
$$\angle OAL=\alpha, \angle OBM=\beta, \angle OCN=\gamma$$
If A, B, C and D are four points such that $$\displaystyle \angle BAC=45^{\circ}\,and\,\angle BDC=45^{\circ}$$ then A, B, C, D are concyclic
In the given triangle, find out $$\angle x$$
If in any triangle the sides a,b,c are respectively 13, 14, 15, then $$r_1$$ = ...., $$r_2$$ = ...., $$r_3$$ = ....