Start with a line segment connecting two points, A and B.  means DA is 3 times longer than BA. Clearly, D cannot fall between A and B because that would mean DA is shorter than BA. So there are two possible locations where D can be placed on the line relative to A and B.
 means DA is 3 times longer than BA. Clearly, D cannot fall between A and B because that would mean DA is shorter than BA. So there are two possible locations where D can be placed on the line relative to A and B.
But with  , or the fact that DB is 2 times longer than BA, we can rule out one of these positions; referring to the attachment, if we place D to the left of A, then DB would be 4 times longer than BA.
, or the fact that DB is 2 times longer than BA, we can rule out one of these positions; referring to the attachment, if we place D to the left of A, then DB would be 4 times longer than BA.
Finally,  , so that CB is 2 times longer than AB. Again we have two possible locations for point C (it cannot fall between A and B), but one of them forces C to occupy the same point as D. However, A, B, C, D are distinct, so C must fall to the left of A.
, so that CB is 2 times longer than AB. Again we have two possible locations for point C (it cannot fall between A and B), but one of them forces C to occupy the same point as D. However, A, B, C, D are distinct, so C must fall to the left of A.
Now let  be the length of AB. Then the length of CD in terms of
 be the length of AB. Then the length of CD in terms of  is
 is  . We have the coordinates of C and D, and the distance between them is
. We have the coordinates of C and D, and the distance between them is  . So
. So

The slope of the line through C and D is

and so the equation of the line through these points is

So the coordinates of A are  . The distance between C and A is
. The distance between C and A is  , so we have
, so we have

Since A falls to the right of C (in the  plane, not just in the sketch), we know to take the positive value
 plane, not just in the sketch), we know to take the positive value  . Then the
. Then the  coordinate is
 coordinate is  .
.
All this to say that A is the point (1, 3), so
