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A(z1),B(z2),C(z3) are the vertices of he...

`A(z_1),B(z_2),C(z_3)` are the vertices of he triangle `A B C` (in anticlockwise). If `/_A B C=pi//4` and `A B=sqrt(2)(B C)` , then prove that `z_2=z_3+i(z_1-z_3)dot`

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Rotating about the point B, we get

`(z_(1)-z_(2))/(z_(3)-z_(2))=(asqrt(2))/(a)e^(ipi//4)`
`=sqrt(2)((1)/(sqrt(2))+(i)/(sqrt(2)))=(1+i)`
or `z_(1)-z_(2)=(z_(3)-z_(2))(1+i)`
or `z_(2)(1-(1+i))=z_(1)-z_(3)(1+i)`
or `z_(2)=(z_(1))/(-i)-(z_(3))/(-i)(1+i)`
`=(iz_(1)-iz_(3)(1+i))=z_(3)+i(z_(1)-z_(3))`
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