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treasure island

Here's the solution that doesn't use analytical geometry:

Call the Coconut C, the Palm P, the cannon N.
Call the coconut point X, the palm point Y, and the treasure point T.

Now, we know XT = YT.  Call this distance d,  Now, draw a line segment that starts at N, is perpendicular to XTY, and has length d,
and ends at a new point I'll call Q.  (Choose the direction so that
angle CNQ = CXT).  Now, CNQ and CXT are congruent triangles (it's basically the same triangle, rotated 90-degrees cround center T).  This means that CT = CQ, and TCQ is a right angle.

Similarly, TYP and QNP are congruent, so TP = PQ, and TPQ is a right angle.

Now, this means that quadrilateral TPQC has two opposite right angles, and the sides next to the right angles are equal.  Gee, I guess that means it has to be a square.  Locating T is now easy.

Wei-Hwa Huang
Thursday, August 15, 2002

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