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Behold!

Author:
Caleb Litster

The Legend

Legend has it that Indian mathematician Bhaskara II entered a room full of other mathematicians, laid a diagram similar to this one on a table, and exclaimed "Behold!" before leaving the room without further comment. The other mathematicians examined the diagram and discovered, to their surprise, that no further words were needed. He had just proved the Pythagorean Theorem!

The Diagram

How did Bhaskara II do it? Without moving any points yet, examine the diagram yourself and make note of any information or relationships you observe.

The Diagram (continued)

Now try adjusting the point E. What do you notice as the figure changes? (Note: the point E is not part of the diagram or proof, it exists simply to aid in dynamic construction.)

Specifically, what do you notice about the shaded areas when E is at the bottom of the large square ()? What about when E is at the top of the large square ()? What about when ?

The Proof

The Pythagorean Theorem says when a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse, they satisfy the relationship . Bhaskara II's diagram is constructed in such a way that all four triangles are right triangles and are congruent, the inner quadrilateral is a square, and the outer quadrilateral is a square.

Attempt to justify algebraically how Bhaskara II's diagram shows that . (Hint: consider how the red, blue, and green shaded areas relate to the area of the large square.)

Reflection

What did you learn from participating in this activity? Is the "Behold!" proof sufficient to convince you the Pythagorean Theorem is true in every case? Does Bhaskara II's proof seem to be missing anything?

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