Complex no. (1 Viewer)

[juantheron]

if are distinct Complex no. and

and satisfy the relation .

then prove that are the vertex of an equilateral

 

[juantheron]

i have tried like in that way



Now

OR



So

Now how can i solve after that

help required...

Thanks

 

[Carrotsticks]

Okay found a proof, but it's gonna take a while to type out.

 

[Carrotsticks]

Okay typing it up messed up and I just couldn't be bothered finding the mistake, so I just wrote it out. I didn't complete all the working out, but the rest is just algebra, and also realising that:



And you should eventually be able to show that:



Which completes the proof that z1, z2 and z3 make an equilateral triangle.

However, the proof below is a bit long with the algebra towards the end. I will try to think of another one when I have time.

The alternative one I have in mind is obtained using the complex roots of unity for:



Then rotating all solutions by some arbitrary angle alpha, and show that it satisfies the given condition ie: z1 z2 and z3 make an equilateral triangle.

I also want to try to find a geometric interpretation of the condition there, so I can somehow relate it to some Circle Geometry theorem, which will facilitate the proof.

 

[Carrotsticks]

Oh and this may be a bit cheap but if you like, you can substitute:



And show that it satisfies the condition, completing the proof.

 

[juantheron]

Thanks Carrotsticks