Show triangles are similar

[Carla1985]

$\text{let } w_1, w_2, w_3\text{ and }z_1,z_2,z_3,\text{ be two triples of distinct points in C.}\\ \text{Suppose f }\in\text{ G such that }f(z_i)=w_i,\text{ for all i=1,2,3.}\\ \text{Show that the triangles with vertices }z_1,z_2,z_3\text{ and }w_1,w_2,w_3\text{ are similar.}$

I have been given this question. I know that it is true but dont know how to go about showing it is true by using translations etc. Could someone point me in the right direction please

 

[Ackbach]

What is $G$?

 

[Carla1985]

Oh sorry, there was a bit earlier on that i missed before some of the questions I already did.

$\text{Let G denote the set of maps C}\mapsto\text{C of the form z}\mapsto\text{az+b for all a,b}\in\text{C. a not equal to 0}$

 

[Ackbach]

Right. So $G$ is a subset of the Mobius transformations, all of which preserve angles. I don't know if you're allowed to assume that or not, but if so, this problem is fairly straight-forward.

 

[Klaas van Aarsen]

$\text{Let G denote the set of maps C}\mapsto\text{C of the form z}\mapsto\text{az+b for all a,b}\in\text{C. a not equal to 0}$

$\text{let } w_1, w_2, w_3\text{ and }z_1,z_2,z_3,\text{ be two triples of distinct points in C.}\\ \text{Suppose f }\in\text{ G such that }f(z_i)=w_i,\text{ for all i=1,2,3.}\\ \text{Show that the triangles with vertices }z_1,z_2,z_3\text{ and }w_1,w_2,w_3\text{ are similar.}$

I have been given this question. I know that it is true but dont know how to go about showing it is true by using translations etc. Could someone point me in the right direction please

Hi Carla1985!

We start with the definition of similarity.
Two triangles are similar if two angles are identical (which means all angles are identical).

So we need the angles.
When we multiply 2 complex numbers, their angles (or more precisely, their so called "arguments") add up.
And when you take the conjugate of a complex number, its angle is negated.
So the angle of 2 complex numbers $p$ and $q$ is the angle of \(\displaystyle p^{}\overline q\).

To get the angle at $z_1$, you need the angle between $(z_2 - z_1)$ and $(z_3 - z_1)$.
Can you find an expression for the angle between them?
And can you also find an expression for the angle between $(w_2 - w_1)$ and $(w_3 - w_1)$?
Or rather $(f(z_2) - f(z_1))$ and $(f(z_3) - f(z_1))$?


Btw, doing so is actually (part of) a proof that Möbius transformations preserve angles, as Ackbach mentioned.