Treasure

Rating: *****
This is one real-world application of advanced mathematics.
Here is the puzzle:
A young man was going through the attic of his grandfather's house and found a paper describing the location of a buried treasure on an particular Island. The note said that on the island one would find a gallows, an oak tree, and a pine tree. To locate the treasure one would begin at the gallows, walk to the pine tree, turn right 90 degrees and walk the same number of paces away from the pine tree. A spike was to be driven at that point. Then return to the gallows, walk to the oak tree and turn left 90 degrees and walk the same number of paces away from the oak tree. Drive a second spike in the ground. The midpoint of a string drawn between the two spikes would locate the treasure.

The young man and his friends mounted an expedition to the island but found the oak tree and the pine tree but no gallows. It had withered years ago without a trace. They returned home with the map below and no treasure.

Can you show them where to look for the treasure?