Hello, John!

Some people found a treasure map, authenticated it and now looking for it.

Here's what it says:

"Start from the tree, walk directly to the falcon-shaped rock.

Count your paces as you walk.

Turn a quarter-circle to the right and walk the same number of paces.

When you reach the end put a stick in the ground.

Return to the tree and likewise walk to the owl-shaped rock, again counting your paces.

Turn a quarter-circle to the left, walk the same number of paces,

and put another stick in the ground. Connect the two sticks with a rope.

The treasure is beneath the midpoint of the rope."

HOWEVER, the tree was no longer on the island.

All that was left was the two weird-shaped rocks.

The people gave up looking for the treasure.

Did the people give up too soon?

Is it possible to find the treasure without the tree? . . . . Yes!

This is a classic problem. Code:

Q
X o
P o |*
o | | *
*| | | *
* | | | *
* | | | *
* | | | *
* | | | *
* | B | | *
F o------*----*-*----*--------o O
* A | C D *
* | *
* | *
o
T

We start at $\displaystyle T$, walk to $\displaystyle F$, turn 90° to the right,

. . and walk to point $\displaystyle P$, where $\displaystyle FP = TF$.

We start at $\displaystyle T$, walk to $\displaystyle O$, turn 90° to the left,

. . and walk to point $\displaystyle Q$, where $\displaystyle OQ = TO$.

The treasure is buried at $\displaystyle X$, the midpoint of $\displaystyle PQ$.

Draw perpendiculars $\displaystyle PA,\:TB,\:XC,\:QD$ to $\displaystyle FO$.

Since $\displaystyle X$ is the midpoint of $\displaystyle PQ$, $\displaystyle XC$ is the __average__ of $\displaystyle PA$ and $\displaystyle QD$.

. . That is: .$\displaystyle XC \:=\:\frac{PA + QD}{2}$ .**[1]**

We find that: .$\displaystyle \Delta PAF \cong \Delta FBT\quad\Rightarrow\quad PA = FB$ .**[2]**

We find that: .$\displaystyle \Delta QDO \cong \Delta OBT\quad\Rightarrow\quad QD = BO$ .**[3]**

Substitute [2] and [3] into [1]: .$\displaystyle XC\:=\:\frac{FB + BO}{2} \:=\:\frac{1}{2}FO$

Hence, the "height" of $\displaystyle X$ is half of the distance between the two rocks.

Since $\displaystyle X$ is the midpoint of $\displaystyle PQ$, it follows that: $\displaystyle C$ is the midpoint of $\displaystyle AD$.

From the congruent right triangles: .$\displaystyle FA = TB$ and $\displaystyle DO = TB$

. . That is, the two end segments of $\displaystyle FO$ are equal.

Hence, $\displaystyle C$ is the midpoint of $\displaystyle FO$.

To locate the treasure $\displaystyle X$, we don't need the tree.

Locate $\displaystyle C$, the midpoint of $\displaystyle FO$.

From $\displaystyle C$, walk perpendicular to $\displaystyle FO$ a distance of $\displaystyle \frac{1}{2}FO$

. . and there is the treasure!