First slide

Geometry of complex numbers

Question

$A (z_{1}), B (z_{2}), C (z_{3})$ are the vertices of a triangle ABC inscribed in the circle $| z | = 2$ . Internal angle bisector of the angle A, meet the circum circle again at $D (z_{4})$ and $2 z_{4}^{2} - k z_{2} z_{3} = 0$ then K =

Difficult

Solution

$B O D = A = C O D$
From rotation about the point ‘O’ we get
$\frac{z_{4}}{z_{2}} = e^{i A}$ , $\frac{z_{3}}{z_{4}} = e^{i A}$
$\Rightarrow z_{4}^{2} = z_{2} z_{3}$

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