By doubling a triangle number we can create a rectangle
I noticed the dimensions were of the form n and n+1 (3 and 4, 4 and 5 and so on)
The dimenesions of the 250th triangle number are 250 and 251
These dimensions give us a rectangle of size 250 x 251 which equals 62750
However, this rectangle size is twice the 250th triangle number so we divide by 2 to get the triangle number
62750 / 2 = 31375
Therefore 31375 is the 250th triangle number

The strategy I use to find any triangle number can be shown by the following
equation I worked out 

t = n(n + 1)
    --------
	   2
	   
The triangle numbers are: 4851, 3655, 7626


The method I used to determine if a number is a triangle number is as follows:

First I double the number to give the rectangle's size
I then get the square root of the number and round it down to give the smaller size of the rectangle - call this n
I then multiply n by n + 1, if this is the same size as the rectangle then this is a triangle number

Yes, it is possible to have a triangle number ending in 000
If we, say, consider the 2000th triangle number 
this would be 2000 x 2001 / 2 which equals 2001000