Is there a formula for the following:
Say a fence cuts across the (right angle) NE corner of a yard (the north line is 51 feet long, the east line is 124 feet long and the fence is 134 feet long; the fence needs to be moved exactly 10 feet further (SW) from where it is. How long do each of the three sides of the triangle need to be? (the north line, the east line and the fence)
I don't think you will find formulas with things like that.
You need define some coordinate system (use the east and north sides as axis) and then find the fence equation.
Then if the fence is dislocated towards SW, this means it is dislocated along a y=x line (considering South and West the positive directions of the axis) and will maintain its slope.
Find the point where that line intersects the fence and calculate the new intersection point in the line after the displacement (point 1 and point 2 should be separated by 10 feet)
After that you get a point and the slope, you can calculate the new equation for the fence and finish it.