Complex Number Question (1 Viewer)

[Aerath]

Any help would be appreciated. =)

The points z₁ z₂ and z₃ are 3 complex numbers that lie on a circle passing though the origin. Prove that the points which represent 1/z₁ , 1/z₂ and 1/z₃ are collinear

 

[azureus88]

Use the fact that exterior angle of cyclic quad is equal to opposite interior angle. You will get:

[maths]arg(z_2-z_3)-arg(z_3)=arg(z_1-z_2)-arg(z_1)\\arg(z_2-z_3)-[arg(z_3)+arg(z_2)]=arg(z_1-z_2)-[arg(z_1)+arg(z_2)]\\arg(\frac{z_2-z_3}{z_2z_3})=arg(\frac{z_1-z_2}{z_1z_2})\\\therefore arg(\frac{1}{z_2}-\frac{1}{z_1})=arg(\frac{1}{z_3}-\frac{1}{z_2})[/maths]

So they are collinear.

 

[Aerath]

Dude, that's genius, thanks.

 

[Drongoski]

Dude, that's genius, thanks.

indeed. congrats !

 

[MC Squidge]

clever stuff

 

[shaon0]

Any help would be appreciated. =)

The points z₁ z₂ and z₃ are 3 complex numbers that lie on a circle passing though the origin. Prove that the points which represent 1/z₁ , 1/z₂ and 1/z₃ are collinear

http://community.boredofstudies.org...0322/foundations-4u-maths-book-s-k-patel.html
- If you scroll to post #8 you will find a solution which is similar to azureus88's solution.