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Need help with complex numbers identity (1 Viewer)

Alasdair09

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Question: "The points A, B, C and O represent the numbers z, 1/z, 1 and 0 respectively. Given that 0<argz<pi/2 prove that the angle <OAC = <OCB"
Please try this question and see if you can find an elegant proof thank you.
 

fan96

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One way is by similar triangles.

We suppose that are distinct points (otherwise the question isn't very well defined).
In particular, . Then,







and we are done.

As a bonus, this doesn't require restrictions on and you also get equalities for the other two angles of the triangles.
 
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Trebla

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Not as elegant but a more intuitive approach is to use the vector rotations.







Similarly





Since the angles lie between zero and 180 degrees then they must be equal.
 

Alasdair09

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One way is by similar triangles.

We suppose that are distinct points (otherwise the question isn't very well defined).
In particular, . Then,







and we are done.

As a bonus, this doesn't require restrictions on and you also get equalities for the other two angles of the triangles.
Thanks for this solution! Although the only thing I do not understand is how you go from |1-1/z|/|z-1| to |1/z||z-1|/|z-1|
 

Drdusk

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Thanks for this solution! Although the only thing I do not understand is how you go from |1-1/z|/|z-1| to |1/z||z-1|/|z-1|
He pulled out a factor of 1/z from the numerator as 1/z * z = 1 and 1/z * 1 = 1/z so like

 

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