Nifty Triangle Geometry

What is math but a formalized set of nifty tricks.

Braver has some challenges whose solutions tickled my fancy.

For example:

Diagonally slice the above parallelograms and rectangle and you end up with a triangle. Turns out every triangle is just a quadrilateral cut in half, and thus the area will simply be half the area of it imaginary whole quadrilateral. The $b \times h$ in $\frac{1}{2}b \times h$ is sister to $l \times b$ or $l \times l$ of quadrilaterals.

Another, is this question:

I was close to getting this but I was blocked in my mind by the focus on only the big square. Turns out, the focus is the area of the big square, which can be seen as:

1. The square of $(a + b)$, which I quickly noticed or
2. (Four times $\frac{1}{2}(a \times b)$) $+$ $c^2$, which is just nice.

Equating these means: $a^2 + 2ab + b^2 = (4 \times \frac {1}{2} (a \times b)) + c^2$

or, $a^2 + 2ab + b^2 = 2ab + c^2$

$\therefore a^2 + b^2 = c^2$

So nifty.

The equivalent of watching Neymar at work.

This was based on his Precalculus Made Difficult book.