Could someone point out some examples which very well displays the problem-solving efficiency of Complex Numbers when it come to Geometrical problems that would otherwise have a long, complex, synthetic solution? They aforementioned problem should not be a 'higher' math one, but rather from elementary Geometry, preferably from some contest.
Please try to show a comparison between the proofs in an understandable manner.
Also, if you could point out some flaws(cases where they don't work, are problematic to work with) of Complex Numbers in Geometry, I would be highly obliged.
Finally, kindly also present some sources for learning contest-level Complex Number Bashing. I know about Evan Chen, but anything else?
Thanks a lot for all solutions/opinions and advices.
One of the nicest example I know deals with the internal maximal area inscribed ellipse named "Steiner ellipse" in a triangle. This ellipse and in particular its focii can be obtained by Marsden's theorem with proof here by reasoning on polynomial $f(z)=(z-a)(z-b)(z-c)$ where $a,b,c \in \mathbb{C}$ are the affixes of the vertices (the foci are the roots of $f'(z)=0$).