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Question Number 195872 by CrispyXYZ last updated on 12/Aug/23

z_1 , z_2 , z_3 ∈C.∣z_1 ∣=∣z_2 ∣=∣z_3 ∣=1. Prove that (((z_1 +z_2 )(z_2 +z_3 )(z_3 +z_1 ))/(z_1 z_2 z_3 ))∈R.

z1,z2,z3∈C.∣z1∣=∣z2∣=∣z3∣=1.Provethat(z1+z2)(z2+z3)(z3+z1)z1z2z3∈R.

Answered by deleteduser1 last updated on 12/Aug/23

=((2z_1 z_2 z_3 +z_1 ^2 z_2 +z_3 ^2 z_1 +z_1 ^2 z_3 +z_2 ^2 z_3 +z_1 z_2 ^2 +z_3 ^2 z_2 )/(z_1 z_2 z_3 )) =(2+(z_1 /z_3 )+(z_3 /z_2 )+(z_1 /z_2 )+(z_2 /z_1 )+(z_2 /z_3 )+(z_3 /z_1 ))=P P∈R⇒P−P^(__) =0⇒2−2+((z_1 z_1 ^− −z_3 z_3 ^− =1−1)/(z_1 z_3 ))+...=0 ■

=2z1z2z3+z12z2+z32z1+z12z3+z22z3+z1z22+z32z2z1z2z3=(2+z1z3+z3z2+z1z2+z2z1+z2z3+z3z1)=PP∈R⇒P−P__=0⇒2−2+z1z−1−z3z−3=1−1z1z3+...=0◼

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