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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

- Thread starter Carla1985
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- Thread starter
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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

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- Jan 26, 2012

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What is $G$?

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- Jan 26, 2012

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- Mar 5, 2012

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

Hi Carla1985!

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

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