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Loci with complex numbers - argument 1

Determine the point of intersection of the two half lines given by the equations below $$ \arg(z - (1 + 2i)) = \frac{\pi}{4},\;\arg(z - (2+i)) = \frac{\pi}{3} $$
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The easiest way to proceed with this question is to establish the cartesian equations of each half line and then determine their intersection by solving a simultaneous equation

The first half-line has equation: $$ \begin{align} \arg(z - (i+2i)) &= \frac{\pi}{4} \\ \tan\arg(z - (i+2i)) &= \tan\frac{\pi}{4} \\ \frac{y-2}{x-1} &= 1 \\ y &= x+1 \end{align} $$

The second half-line has equation: $$ \begin{align} \arg(z - (2+i)) &= \frac{\pi}{3} \\ \tan\arg(z - (2+i)) &= \tan\frac{\pi}{3} \\ \frac{y-1}{x-2} &= \sqrt{3} \\ y &= \sqrt{3}x - 2\sqrt{3} + 1 \end{align} $$

Solving this pair of equations gives $$z = \frac{2\sqrt{3}}{\sqrt{3}-1} + \frac{3\sqrt{3}-1}{\sqrt{3}-1}i$$