Originally Posted by mr fantastic

If you don't know the coordinates of the point outside of the square, then one solution is to draw the four different triangles formed by two segments and a side of the square. In each triangle you know the length of a side (edge length of square = 1) and the angle between the two unknown sides (the angle between two segments). You can then use the cosine rule to write down four equations, each equation will have the length of two segments in it.

Four equations (cosine rule applied to each triangle) and four unknowns (four segments). Solve simultaneously.