∠t in triangle utx **is at most** !

120 **) now** 
according to Law of cosines in triangles( c a b 2abcosC
2 2 2
= + − **), in triangle** 
 utx **we have** 
ux = ut + xt − 2ut * xt *cos∠utx 2 2 2 also we know Rs ut = xt = , and, ∠utx ≤120 , 

```
and cosine is a decreasing function in [ ]
                                              !
                                               
                                        ,0 180 so we 
have 
ux = Rs + Rs − 2Rs*cos∠utx ≤
    2 2 2 2 
    2 2 2
2Rs − 2Rs *cos120 = 3Rs
                         !
                          

```

So we have ux ≤ 3 *Rs and this contradict with our assumption that 2 + 3 *Rs ≤ Rc and R ux c ≤ **. Also we know that** 3 < 2 + 3 . 

So we have Rs ts < **, now we prove that at least one** 
of the points s or t **are inside region** A . 

![0_image_2.png](0_image_2.png)

For this, first we assume that both points s and t **are** 
outside the region A **, so because the intersection of** 
ts and ux is inside the region A **(the region** A **is** 
convex and both points u and x **are inside region** 
A and so all of the points of ux **inside region** A )line ts **intersect with two sides of region** A , 
these two side can't be two parallel sides of region A because in this case the length of ts **should be** 
more than the length of the other sides of region A **and this can't be true because we assume that** 
A = a *b **is a rectangular region such that** 
Rs ≤ a,Rs ≤ b **, also we prove before** Rs ts < **. So** 
ts intersect with two joint sides of region A **(Figure** 
2.) in this case we have ≤ ∠sEt ≤ ∠sut !

90
(points u and x **are in two different sides of line** ts and we assume that u **is in the side that restricted by** 
two joint sides of the rectangle and line ts **.) and this** 
can't be true because we prove before Rs ts < **and** 
that imply ! ∠sut ≤ 60 . 

So at least one of the points s or t **are inside** region A , we assume that, point t **is inside region** A . 

So the point t **is at least 1-covered, we name the node** that covers t as y **(Figure 3).** 

![0_image_0.png](0_image_0.png)

![0_image_1.png](0_image_1.png)

Now we prove that uy < Rc and Rc yx < **, for this** 
first we assume that R uy c ≤ **, in this case the angle** 
∠ytu **must be greater than or equal to** !

150 **because** 
we have ut = Rs and Rs yt < and R uy c ≤ **, now** 
according to law of cosines in triangles( uty **) , we** 
have uy = ut + yt − 2ut * yt *cos∠ytu 2 2 2 Now if the angle ∠ytu **is less than** !

90 **then it's** 
cosine is positive and we have uy < 2Rs uy < 2 * Rs < 2 + 3 * Rs = Rc 2 2 And if the angle ∠ytu **is greater than** !

90 **then it's** 
cosine is negative and we have uy R R ytu < 2 s − 2 s
*cos∠
2 2 2**, if the angle** 

```
∠ytu is less than !
        150 since cosine is a decreasing 
function in [ ]
        !
        
     ,0 180 we have 
                       !
                        
2 2 *cos 2 2 *cos150 2 2 2 2
s s Rs Rs R − R ∠ytu < −

```

= 2( + *)3 Rs uy < 2 + 3 *Rs = Rc 2, and this can't be true because we assume that R uy c ≤ , so the angle ∠ytu **is greater than or** 