Geometry Puzzle - Solution

Solution: It looks like a triangle, because a thick line was used. The hypotenuse of the composite triangle is actually not a straight line – it is made of two lines.

The "64 = 65 paradox" arises from the fact that the edges of the four pieces, which lie along the diagonal of the formed rectangle, do not coincide exactly in direction.
This diagonal is not a straight segment line but a small lozenge (diamond-shaped figure), whose acute angle is arctan 2/3 - arctan 3/8 = arctan 1/46 which is less than 1 degree 15' . Only a very precise drawing can enable us to distinguish such a small angle. Using analytic geometry or trigonometry, we can easily prove that the area of the "hidden" lozenge is equal to that of a small square of the chessboard.

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