Argument d'un quotient

P roposition

Soit \(\text A(z_\text A)\) , \(\text B(z_\text B)\) et \(\text C(z_\text C)\) trois points distincts du plan complexe. On a : \(\begin{align*}\arg\left(\frac{z_\text C-z_\text A}{z_\text B-z_\text A}\right) \equiv \left(\overrightarrow{\text A\text B};\overrightarrow{\text A\text C}\right) \ [2\pi].\end{align*}\)

Démonstration

On a   \(\begin{align*}\arg\left(\frac{z_\text C-z_\text A}{z_\text B-z_\text A}\right)& \equiv \arg\left(z_\text C-z_\text A\right)-\arg\left(z_\text B-z_\text A\right) \ [2\pi]\end{align*}\)
avec \(\arg\left(z_\text C-z_\text A\right) \equiv \left(\vec{u};\overrightarrow{\text A\text C}\right) \ [2\pi]\)  et \(\arg\left(z_\text B-z_\text A\right) \equiv \left(\vec{u};\overrightarrow{\text A\text B}\right) \ [2\pi]\) .
Ainsi
\(\begin{align*}\arg\left(\frac{z_\text C-z_\text A}{z_\text B-z_\text A}\right)& \equiv \left(\vec{u};\overrightarrow{\text A\text C}\right)-\left(\vec{u};\overrightarrow{\text A\text B}\right) \ [2\pi]\\& \equiv \left(\vec{u};\overrightarrow{\text A\text C}\right)+\left(\overrightarrow{\text A\text B};\vec{u}\right) \ [2\pi]\\& \equiv \left(\overrightarrow{\text A\text B};\vec{u}\right)+\left(\vec{u};\overrightarrow{\text A\text C}\right) \ [2\pi]\\& \equiv \left(\overrightarrow{\text A\text B};\overrightarrow{\text A\text C}\right) \ [2\pi].\end{align*}\)