### Solution of Triangle

goals
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 $r_1$ = ...., $r_2$ = ...., $r_3$ = ....