Geometric interpretations of an equality.

Question

I need to prove that for complex numbers $w_1, w_2$ and $w_3$ if: $$\frac{w_2-w_1}{w_3-w_1}=\frac{w_1-w_3}{w_2-w_3}$$ then: $$|w_2-w_1|=|w_3-w_1|=|w_2-w_3|$$ by geometric interpretation of the given equality. Thanks.

Answer 1

If you are allowed to do just a little bit of algebra to start with you could proceed as follows. Taking the argument of both sides, $$\arg(w_2-w_1)-\arg(w_3-w_1)=\arg(w_1-w_3)-\arg(w_2-w_3)\ .$$ This means, subject to being careful about the non-uniqueness of the argument that the angles $$\angle w_3.w_1.w_2\quad\hbox{and}\quad \angle w_2.w_3.w_1$$ are equal. Also, if (real or complex) fractions $a/b$ and $c/d$ are equal, then they are also equal to $(a+c)/(b+d)$, provided $b+d\ne0$. Hence we also have $$\frac{w_2-w_1}{w_3-w_1}=\frac{w_3-w_2}{w_1-w_2}\ ,$$ and by the same argument, $$\angle w_1.w_2.w_3$$ is the same as above. Therefore $w_1,w_2,w_3$ form an equilateral triangle, and your conclusion follows.

This answer probably has more algebra than you would like, but the geometrical point is that you can, in effect, interpret the given quotients as angles.