Actually my questions below are somewhat similar to geometric proofs. But it's more focused on application of a vector in the complex plane.
The question :
In parallelogram $ABCD$, point $E$ bisects side $AD$. Prove that the point where $BE$ meets $AC$ trisects $AC$.
In geometric, we usually use the previous theorem, lemma, or definition to prove this problem. But when we're dealing with complex numbers which have vector representation of them, we don't need those theorem, lemma and definition anymore, do we? Then what is the idea of this problem. Please give me a clue, and i will prove it by my self.
Thanks for advance.
Descartes discovered what we now call "analytic geometry". The essential idea is that the theorems you could prove with (synthetic) geometric arguments like those in Euclid you could also prove by working algebraically with coordinates of points. Later on, when others realized that the complex numbers could be thought of as points in the plane, the algebra of complex numbers turned out to be another way to prove those theorems.
Which kind of proof to use depends on your taste, your abilities, and the theorems you care about. Sometimes one proof offers more intuitive content or a more general argument than the other.
The particular theorem you ask about can be proved with a routine calculation using complex numbers, but I find the synthetic proof that sees similar triangles one twice the other much more beautiful and much more informative.
I think we sometimes still