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 circumcircle again at $D (z_{4})$ then:

Moderate

A

$z_{4}^{2} = z_{2} z_{3}$
B

$z_{4} = \frac{z_{2} z_{3}}{z_{1}}$
C

$z_{4} = \frac{z_{1} z_{2}}{z_{3}}$
D

$z_{4} = \frac{z_{1} z_{3}}{z_{2}}$

Solution

$∠ B A D = ∠ C A D$ $\Rightarrow a r c B D = a r c C D$
$\Rightarrow$ $∠ B O D = ∠ C O D = A$
$\frac{z_{4}}{z_{2}} = e^{i A} = \frac{z_{3}}{z_{4}}$
$\Rightarrow z_{4}^{2} = z_{2} z_{3}$

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