<s>[INST] Pythagorean Theorem for imaginary numbers If we let one leg be real-valued and the other leg equal $bi$ then the Pythagorean Theorem changes to $a^2-b^2=c^2$ which results in some kooky numbers.
For what reason does this not make sense? Does the Theorem only work on real numbers? Why not imaginary?
 [/INST] In the vein of @AndréNicolas' comment that "ideas like this can be useful" ...
Heron's formula for the area, $W$, of a (non-obtuse) triangle in terms of the lengths of its sides is equivalent to the Pythagorean Theorem for Right-Corner Tetrahedra:
$$W^2 = X^2 + Y^2 + Z^2$$
where $W$ is the "hypotenuse-face" opposite three mutually-perpendicular edges that form right triangles with areas $X$, $Y$, $Z$.
Specifically, if we position the right corner at the origin in 3-space, choose the other vertices at $(x,0,0)$, $(0,y,0)$, $(0,0,z)$, and write
$$\begin{align}
a^2 = y^2 + z^2 \qquad b^2 &= z^2 + x^2 \qquad c^2 = x^2 + y^2 \\
X = \frac{1}{2} y z \qquad Y &= \frac{1}{2} z x \qquad Z = \frac{1}{2} x y
\end{align}$$
then 
$$x^2 = \frac{1}{2}\left(-a^2 + b^2 + c^2\right) \qquad y^2 = \frac{1}{2}\left(a^2-b^2+c^2\right) \qquad z^2 = \frac{1}{2}\left(a^2+b^2-c^2\right)$$
whence
$$\begin{align}
W^2 = X^2 + Y^2 + Z^2 &= \frac{1}{4}\left(y^2 z^2+z^2x^2+x^2y^2\right)\\
&=\frac{1}{16}\left(-a^4-b^4-c^4+2a^2b^2+2b^2c^2+2c^2a^2\right) \\
&=\frac{1}{16}\left(a+b+c\right)\left(-a+b+c\right)\left(a-b+c\right)\left(a+b-c\right)
\end{align}$$
which is Heron's Formula for $W$ in terms of side-lengths $a$, $b$, $c$.
Note that I wrote the equivalence is for "non-obtuse" triangles. This is because "hypotenuse-face" $W$ cannot have an obtuse angle; to see this, write $\theta$ for the angle opposite side $a$, so that
$$\cos \theta =\frac{-a^2+b^2+c^2}{2b c} = \frac{x^2}{bc}$$
This value is non-negative (and, hence, $\theta$ is non-obtuse) for any edge-length $x$ ... well, any real edge-length $x$. However, if we allow $x$ to be an imaginary length, then $\theta$ can be obtuse, and then Heron's Formula can be seen to apply to all triangles.
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