The warden of the local prison was in a good mood - after a long winter, it was finally spring. The weather was warm and the sun shined over the prison. The warden decided, as is customary in the state of Puzzlevania, to offer the prisoners a chance to escape.
Her prisoners were very clever, and she decided to pose to them an equally clever problem. She found a suitable location for her challenge, where the ground was perfectly flat and free of grass. There was only one structure nearby - one tall chain-link fence, which lay to the east, whose top ran in a perfectly straight line. In her flat area, she placed four pegs in a square. The diagonals of the square measured $10$ feet and one of the diagonals was parallel to the top of the prison wall.
Around each peg, she placed a glass dome, such that the prisoners could not touch the pegs. She took care to affix the domes to the ground firmly, such that the prisoners could not possibly move or break them. The domes were of no precise radius and were not centered around the pegs. They did not distort the light at all and the prisoners could see through them clearly.
She brought the prisoners out to the yard, and showed them the setup and declared:
As you can see, I have placed four pegs into the ground here, in a square. If you all can devise and execute a scheme to determine the exact center of the square, I will set you all free. However, by the canonical laws of Puzzlevania, I will execute all of you if you are even the slightest bit off.
To accomplish this task, she gave them a piece of wood, with one perfectly straight edge at least $10$ feet long. They have access to no other tools. The prisoners took this information back to the cafeteria and drew out the situation. They produced the following image:
where the thick line to the right (east) is the prison wall, the points $A$, $B$, $C$ and $D$ are the pegs, and $P$ is the point they must find, at the center of the square. The dashed circles are the diagonals of the squares and $AC$ is perpendicular to the wall where $BD$ is parallel to the wall.
The prisoners, having been master carpenters and geometers, have the ability to hold the wood wherever they want with perfect precision. They can see the centers of the pegs perfectly clearly. They cannot make any marks - rather, they must, at some point in time, be able to direct the warden exactly to the center.
How can they find the square's center and be released?