TITLE: Japanese Temple Problem From 1844
QUESTION [71 upvotes]: I recently learnt a Japanese geometry temple problem. 
The problem is the following:
Five squares are arranged as the image shows. Prove that the area of triangle T and the area of  square S are equal.

This is problem 6 in this article.
I am thinking about law of cosines, but I have not been able to prove the theorem. Any hints would be appreciated.

REPLY [12 votes]: The four triangles adjacent to $S$ (two of them right, two of them obtuse) all have the same area.  (Each has the same base and height as the one on the opposite side of the square, while the two right triangles are congruent.). Now rotate each of the obtuse triangles by $90^\circ$ so that they are adjacent to $T$, as shown.

What now needs to be proved is that the two shaded pentagons have equal area.  This can be done by observing that each pentagon decomposes into an isosceles right triangle and a trapezoid.  The isosceles right triangles are congruent; the trapezoids have equal area as their two bases are the same and their heights are the same.