1.

For three non-colliner complex numbers `Z,Z_(1)` and `Z_(2)` prove that `|Z-(Z_(1)+z_(2))/(2)|^(2) + |(Z_(1) -Z_(2))/(2)|= (1)/(2) | Z - Z_(1)|^(2) +(1)/(2)|Z- Z_(2)|^(2)`

Answer» Consider the formed by A(Z), `B(Z_(1))` and `C(Z_(2))`.
Let midpoint of BC be D having complex number `(Z_(1)+Z_(2))/(2)`
By Apollononius therorem, we have
`AB^(2) + AC^(2) + 2(AD^(2) + BD^(2))`
`therefore |Z-(Z_(1) +Z_(2))/(2)|^(2) + |(Z_(1) +Z_(2))/(2)|`
`AD^(2) + BD^(2)`
`=(1)/(2)AB^(2) + (1)/(2)Ac^(2)`
`= (1)/(2)|Z- Z_(1)|^(2)+(1)/(2)|Z-Z_(2)|^(2)`


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