I should have depicted the following tetrahedron using tikz of tex. The image below is from this instagram post
$$ \color{fuchsia}{\underbrace{S_4^2= S_1^2+S_2^2+S_3^2}_{\text{I want to derive this} } } $$
$~ S_i ~$ represents the ith area of a triangle.
What I've done for deriving it are as follows.
$$\begin{cases}
(0,0,0) \leftarrow~~\text{coordinates where the 3 right angle symbols gather}\\
(a,0,0) \leftarrow~~\text{coordinates on x-axis}\\
(0,b,0) \leftarrow~~\text{coordinates on y-axis}\\
(0,0,c) \leftarrow~~\text{coordinates on z-axis}\\
\end{cases}$$
Please assume that x-axis is along with left and y-axis is along with right and z-axis with vertical.
$$\begin{cases}
S_1={1 \over 2 }bc\\
S_2={1 \over 2 }ac\\
S_3={1 \over 2 }ab\\
\end{cases}$$
So the remaining problem is to find out the formula of $~ S_4 ~$
What I've thought for it is to firstly find out the perpendicular line between $~ \text{AB} ~$ and $~ \text{C} ~$.
If that is done, then the coordinates on $~\text{AB} ~$ which is passed by that perpendicular line is known(we define this position as $~ \text{D} ~$ )
Hence
$$ S_4={1 \over 2 } \left( \text{AD} \cdot \text{CD} + \text{BD} \cdot \text{CD} \right) $$
But this way seems a bit complicated one I think.
Is there some more simpler way to derive the area of $~ S_4 ~$ ?
In the first place I don't know how to find the perpendicular line…
Best Answer
It is not necessary to derive an exact formula of a length of the perpendicular line. That is the key factor for proving the squared pink eqn.
$$\begin{align} p&:=\text{length of perpendicular line betwneen C and AB} \\ q&:=\text{length of line segment between the origin and AB} \\ d&:=\text{length of AB} \\ S_4^2&= \left({1 \over 2 }pd \right)^2\\&= {p^2 d^2 \over 4 }\iff 4S_4^2=p^2d^2\\ 4S_4^2&=p^2d^2\\ &=d^2 \left(c^2+q^2 \right)\\ &=c^2d^2+d^2q^2\\ &=c^2 \left(a^2+b^2 \right)+d^2q^2\\ &=\cdots\\ &=4S_1^2+4S_2^2+4S_3^2\\ \therefore S_4^2&=S_1^2+S_2^2+S_3^2 \end{align}$$
BTW using tikz to depict a figure is very time taking for me so far. But should I still use it? I wish I can use a tool which draw a figure with realtime response.