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 , walk to , turn 90° to the right,

. . and walk to point , where .

We start at , walk to , turn 90° to the left,

. . and walk to point , where .

The treasure is buried at , the midpoint of .

Draw perpendiculars to .

Since is the midpoint of , is the __average__ of and .

. . That is: . .**[1]**

We find that: . .**[2]**

We find that: . .**[3]**

Substitute [2] and [3] into [1]: .

Hence, the "height" of is half of the distance between the two rocks.

Since is the midpoint of , it follows that: is the midpoint of .

From the congruent right triangles: . and

. . That is, the two end segments of are equal.

Hence, is the midpoint of .

To locate the treasure , we don't need the tree.

Locate , the midpoint of .

From , walk perpendicular to a distance of

. . and there is the treasure!