We have a triangle ABC on the complex plane whose vertices are the three points $\mathrm { A } ( \alpha )$, $\mathrm { B } ( \beta )$ and $\mathrm { C } ( \gamma )$ that satisfy $$\frac { \gamma - \alpha } { \beta - \alpha } = 1 - i$$ (In the following, the range of an argument $\theta$ is $0 \leqq \theta < 2 \pi$.) (1) When we express the complex number $\frac { \gamma - \alpha } { \beta - \alpha }$ in polar form, we have $$\frac { \gamma - \alpha } { \beta - \alpha } = \sqrt { \mathbf { N } } \left( \cos \frac { \mathbf { O } } { \mathbf { P } } \pi + i \sin \frac { \mathbf { O } } { \mathbf { P } } \pi \right) .$$ Hence we see that point C is the point resulting from rotating point B by $\frac { \square \mathbf { Q } } { \mathbf{R} } \pi$ around point A and then changing its distance from point A to its distance multiplied by $\sqrt { \mathbf { S } }$. From this we also see that the absolute value and the argument of the complex number $w = \frac { \gamma - \beta } { \alpha - \beta }$ are $$| w | = \mathbf { T } \quad \text { and } \quad \arg w = \frac { \mathbf { U } } { \mathbf { 4 } } \pi .$$ (2) If $\alpha + \beta + \gamma = 0$, then we have that $$| \alpha | : | \beta | : | \gamma | = \sqrt { \mathbf { W } } : \sqrt { \mathbf { X } } : \sqrt { \mathbf { Y } } .$$
We have a triangle ABC on the complex plane whose vertices are the three points $\mathrm { A } ( \alpha )$, $\mathrm { B } ( \beta )$ and $\mathrm { C } ( \gamma )$ that satisfy
$$\frac { \gamma - \alpha } { \beta - \alpha } = 1 - i$$
(In the following, the range of an argument $\theta$ is $0 \leqq \theta < 2 \pi$.)\\
(1) When we express the complex number $\frac { \gamma - \alpha } { \beta - \alpha }$ in polar form, we have
$$\frac { \gamma - \alpha } { \beta - \alpha } = \sqrt { \mathbf { N } } \left( \cos \frac { \mathbf { O } } { \mathbf { P } } \pi + i \sin \frac { \mathbf { O } } { \mathbf { P } } \pi \right) .$$
Hence we see that point C is the point resulting from rotating point B by $\frac { \square \mathbf { Q } } { \mathbf{R} } \pi$ around point A and then changing its distance from point A to its distance multiplied by $\sqrt { \mathbf { S } }$. From this we also see that the absolute value and the argument of the complex number $w = \frac { \gamma - \beta } { \alpha - \beta }$ are
$$| w | = \mathbf { T } \quad \text { and } \quad \arg w = \frac { \mathbf { U } } { \mathbf { 4 } } \pi .$$
(2) If $\alpha + \beta + \gamma = 0$, then we have that
$$| \alpha | : | \beta | : | \gamma | = \sqrt { \mathbf { W } } : \sqrt { \mathbf { X } } : \sqrt { \mathbf { Y } } .$$