James Garfield's Pythagorean Proof

Today I learned James Garfield, who once worked as a lawyer, Civil War General, and served as the 20th President of the United States, was math savvy and published a Pythagorean theorem proof.^[1]

First, we can find the area of a trapezoid with the following equation:

\begin{align*} \frac{\displaylines{(a+b)^2}}2 \end{align*}

We can find the area of each outer triangle with the following:

\begin{align*} \frac{\displaylines{ab}}2 \end{align*}

And the area of the inner triangle with:

\begin{align*} \frac{\displaylines{c^2}}2 \end{align*}

Reducing, we can go to the end:

\begin{align*} \displaystyle{\displaylines{\frac{(a+b)^2}2}} =\displaystyle{\displaylines{2\frac{ab}2+\frac{c^2}2}} \end{align*} \begin{align*} \displaystyle{\displaylines{\frac{a^2+2ab+b^2}2 = ab+\frac{c^2}2}} \end{align*} \begin{align*} \displaystyle{\displaylines{a^2+2ab+b^2 = 2ab+c^2}} \end{align*} \begin{align*} \displaystyle{\displaylines{a^2+b^2=c^2}} \end{align*}