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• Pythagorean Theorem | Proof

Pythagorean Theorem | Proof

In this paper, I prove the pythagorean theorem. Formally, the pythagorean theorem is all about right triangles. So, for any right triangle with sides $a, b$, and $c$ where $a, b$ are the legs and $c$ is the hypotenuse. We know by the pythagorean theorem that $a^2 + b^2 = c^2$. Our goal is to prove this fact. Consider a square made from the sum of the legs $a+b$. We know the area would be the square of this term, which is $(a + b)^2$, expanding it gives $\ell_0 = (a + b)^2 = a^2 + b^2 + 2ab$. This is the area of the square, we know this as true. Now if we join the points between $a$ and $b$ together along all the sides, we end up with 4 similar right triangles inside the square. Alongside this, we create a 4-figure in the interior. In the triangles, we know that two legs are congruent (sides $a$ and $b$ respectively) and the interior angle is a right angle amongst all triangles. This is sufficient to show that all 4 triangles are congruent in the square. The third side, then, must be the same amongst all the triangles, we will call it $c$. Now, since all the sides of the 4-figure are equal and one angle is right (this is due to the fact that the complementary angles of a right triangle sum to $\pi/2$), we know that the 4-figure is a square. In terms of area then, if we break up the big square into the sum of the interior square and 4 congruent right triangles, we get $\ell_1 = 2ab + c^2$ as the area as well. But, both forms of the area $\ell_0$ and $\ell_1$ are the same since they are about the same object. So, we get…

which gives the pythagorean theorem, as desired. $\blacksquare$