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Second Proof of Pythagorean Theorem by Shear Transformation

Author:Brad Ballinger
In the animation above, the green and purple quadrilaterals go through three stages, which I call "square", "parallelogram", and "rectangle". I hope this isn't confusing; after all, every square is a rectangle, and every rectangle is a parallelogram. When I say "rectangle" in these questions, I mean the non-square ones; and when I say "parallelogram", I mean the non-rectangular ones that have an edge along the line drawn through C.

Question 1

At three times in the animation, Triangle ABC is rotated by some amount. What are the measures of the three angles of rotation?

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Question 2

The green square is transformed into a parallelogram. What reason is there to think that this parallelogram has the same area as the original green square?

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Sub-question 2a

Make sure C is closer to B than to A. Pause the animation while the purple square is being transformed into the purple parallelogram. At this moment, how can we know that the two green triangles have the same area?

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Question 3

The green and purple parallelograms are transformed together into a pair of adjacent rectangles. How do we know that these rectangles have the same areas as the matching parallelograms they came from?

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Question 4

How do we know the green and purple rectangles combine to make a square, and that this square has the same area as the orange square?

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