I am stuck with the following problem that says
If the complex numbers $z_1,z_2,z_3$ represents the three points $P,Q, R $ respectively and be such that $lz_1+mz_2+nz_3=0$ where $ l+m+n=0 $ then show that $P,Q, R$ lie on a straight line.
Here is my try: $lz_1+mz_2+nz_3=0 \implies -(m+n)z_1+mz_2+nz_3=0$ which gives $$m(z_2-z_1)=n(z_1-z_3) \tag1$$
Similarly, $lz_1+mz_2+nz_3=0 \implies -(m+l)z_3+mz_2+lz_1=0$ which gives $$m(z_3-z_2)=l(z_1-z_3) \tag2$$
Now from (1) and (2) we get ,$$ln(z_1-z_3)=lm(z_2-z_1)=mn(z_3-z_2)$$
Now, what next? Can someone help?
Best Answer
It's wrong for $l=m=n=0$.
If $m\neq0$ and $n\neq0$ so after writing $$m(z_2-z_1)=n(z_1-z_3)$$ we are done because it's just $$\vec{PQ}||\vec{PR}.$$