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Complex numbe...

Question

Complex numbers $z_{1}, z_{2}, z_{3}$ are the vertices $A, B, C$ respectively of an isosceles right angled with right angle at $C$ . Show that $(z_{1} - z_{2})^{2} = 2 (z_{1} - z_{3}) (z_{3} - z_{2})$

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Solution

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Applying rotation about point C
$\frac{z_{1} - z_{3}}{z_{1} - z_{3}} = e^{i \frac{π}{2}} . . . (i)$
Applying rotation about point B
$\frac{z_{1} - z_{2}}{z_{3} - z_{2}} = \sqrt{2} e^{\frac{i π}{4}} . . . (i i)$
Applying rotation about point A
$\frac{z_{2} - z_{1}}{z_{3} - z_{1}} = \sqrt{2} e^{- i \frac{π}{4}} . . . (i i i)$
Multiplying (ii) and (iii) $\frac{(z_{1} - z_{2}) (z_{2} - z_{1})}{(z_{3} - z_{2}) (z_{3} - z_{1})} = 2$
By solving $(z_{1} - z_{2})^{2} = 2 (z_{1} - z_{3}) (z_{3} - z_{2})$

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