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 eliminated years ago without
a trace. They returned home with the map below and no treasure.

Show them where to look for the treasure.

Try assuming an arbitrary location for the gallows and use GSP to construct the
location of the two spikes. Now move the location of the gallows. What happens?
Why?

Click **here**
for a GSP file with the map of the Island. Use the GSP file to construct the location of the
treasure for an arbitrary point on the Island for the gallows. Then drag the point for the
gallows to a different location.
Observatons?

HINT for Geometric Proof: For an arbitrary point for the
Gallows, construct right triangles to represent the directions. Construct the line determined by
the two trees and drop perpendiculars to that line from the gallows and from
each of the stakes at the other acute vertices of the right triangles.

See also: **Isosceles
Right Triangles--Path of the Mid-Point **

See Gamow, George
(1947). ** One Two Three
. . . Infinity: Facts and
Speculations of Science** for a presentation of one version of
this problem. Gamow’s
solution uses a presentation of complex numbers. Click

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