Using CAT(0) inequality Let $X$ be a CAT(0) space with metric $d$. Let $p,x,y$ three points on $X$, and let $u,v$ be points on geodesic $[p,x]$ and geodesic $[p,y]$ such that $d(p,u)\geq a,d(p,v)\geq a$,where a is some positive constant. Now I got bit confuse to show by using CAT(0) inequality that 
$d(u,v)\geq (a) \sin(\angle_p(x,y)/2) $. 
Any help appreciated. For any details please refer to 
http://www.math.bgu.ac.il/~barakw/rigidity/bh.pdf 
I am trying to understand the proof of Prop 1.4 page 400 

This is just an application of the law of sines:
$$\frac{d(u,v)}{(\rho - \delta)} \geq \frac{d(u,v)}{d(p,v)} \geq \frac{d(u,v)}{2 d(p,v)} \geq \sin(\frac{\angle_p(x,y)}{2})$$
