The Pythagorean theorem and the Euclidean plane
The pattern of the blue square ABJK tiles the 2-d plane.
The pattern of the red square CBFG and the green square ADEC also tiles the 2-d plane.
Without resorting to cutting up the squares, show that the homogenity of the Euclidean space is the necessary and sufficient reason for the area of the blue square to be equal the combined area of the red and green squares.