In the complex plane ,the vertices of an equilateral triangle are represented by the complex numbers #z_1,z_2# and #z_3# ,prove that #1/(z_1-z_2)+1/(z_2-z_3)+1/(z_3-z_1)=0#?

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#1/(z_1-z_2)+1/(z_2-z_3)+1/(z_3-z_1)=0#

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Answer 1 · 2018-01-03T16:59:09

See below.

Calling #u = z_1-z_2# with #z_1,z_2# vertices of an equilateral triangle, the other two sides can be represented as

#v = u e^(i phi)#
#w = u e^(2i phi)# with #phi =+-2/3pi# (triangle can be reflected)

Now,

#1/u+1/v+1/w = 1/(z_1-z_2)(1+e^(-iphi) + e^(-2i phi)) = 1/(z_1-z_2)((e^(-3i phi)-1)/(e^(-i phi) -1))#

but the numerator

#e^(-3iphi)-1 = e^(+-3i 2/3pi)-1 = 1 -1 = 0# then finally

#1/u+1/v+1/w =0#