One very famous proof of the Pythagorean Theorem is by the Hindu mathematician Bhaskara. It

is often called the Behold proof because, as the
story goes, Bhaskara drew the figure below and
offered no verbal argument other than to exclaim,
”Behold.” Use algebra to fill in the steps, explaining why this figure proves the Pythagorean Theorem.
to find image just search for "behold proof square image"

I like what Sal Khan did with it.

https://www.khanacademy.org/math/basic-geo/basic-geometry-pythagorean-theorem/basic-geometry-pythagorean-proofs/v/bhaskara-s-proof-of-pythagorean-theorem-avi

The Behold proof of the Pythagorean Theorem can be explained using algebraic steps. Here is how it works:

Step 1: Start with a right triangle, where the two legs have lengths a and b, and the hypotenuse has length c.

Step 2: Draw squares on each side of the triangle, where the areas of the squares are a^2, b^2, and c^2.

Step 3: Divide the larger square (c^2) into two rectangles as shown in the figure.

Step 4: The area of the larger square (c^2) can be expressed as the sum of the areas of the two smaller squares (a^2 + b^2), plus the area of the two rectangles (2ab).

Step 5: Algebrically, this relationship can be written as c^2 = a^2 + b^2 + 2ab.

Step 6: Rearrange the equation to isolate the terms involving 'a' and 'b' together: c^2 = (a^2 + 2ab + b^2).

Step 7: Recognize that (a + b)^2 can be written as a^2 + 2ab + b^2.

Step 8: Therefore, the equation can be rewritten as c^2 = (a + b)^2.

Step 9: Taking the square root of both sides, we have c = (a + b).

Step 10: This proves that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs, which is the Pythagorean Theorem.

So, by using algebraic steps, the Behold proof demonstrates why the figure represents the Pythagorean Theorem.

To understand why the figure in the "behold" proof represents the Pythagorean Theorem, let's break it down step by step using algebra.

Step 1: Start with a right-angled triangle. Draw a square on each of the three sides of the triangle.

Step 2: Label the lengths of the sides. Let's call the lengths of the two shorter sides a and b, and the hypotenuse (the longest side) c.

Step 3: In the figure, we can see that the area of the larger square, which represents c^2, is composed of four smaller squares (a^2, b^2, a^2, b^2).

Step 4: Let's calculate the areas of these individual squares. The area of the first smaller square is a^2, the area of the second smaller square is b^2, the area of the third smaller square is also a^2, and the area of the fourth smaller square is b^2.

Step 5: Now, consider the smaller squares on the sides. The total area of the two smaller squares on the legs of the triangle (a^2 + b^2) is equal to the area of the larger square (c^2).

Step 6: This means that a^2 + b^2 = c^2, which is the Pythagorean Theorem. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Therefore, the figure in the "behold" proof visually represents the Pythagorean Theorem by showing the relationship between the squares on each side of a right-angled triangle.