# Consider a Quadrilateral PQRS as given in the diagram with paths AB,BC and CD as shown. Then what is the minimum value of $(AB+BC+CD)^2 -20$? I tried using coordinate geometry to find the distance AB,BC and CD and then finding their minimum by partially differentiating the equations. This yielded a rather complicated equation which I could not solve.

I'm sure there must be a more direct and simpler approach to this question, but I am simply hitting dead ends.

Any hints on how to solve the question?

Your given problem is to compute the shortest path from $$A$$ to $$D$$, given the requirement that the path must touch the lines $$QR$$ and $$PS$$.
Consider a slightly different problem: let $$A'$$ be the reflection of $$A$$ about the line $$QR$$, and let $$D'$$ be the reflection of $$D$$ about the line $$PS$$. What is the shortest path from $$A'$$ to $$D'$$, given the requirement that the path must touch the lines $$QR$$ and $$PS$$? Can you solve this problem? Are the two problems related in any way?