if z1 z2 z3 z4 be the vertices of a square in the argand plane, then prove that 2z2=(1-i)z1+(1+i)z3 - Maths - Complex Numbers and Quadratic Equations

Question

if z1 z2 z3 z4 be the vertices of a square in the argand plane,
then prove that 2z2=(1-i)z1+(1+i)z3

Priyanka Kedia · Accepted Answer

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Please find below the solution to the asked query:

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Given, z1,z2,z3and z4 are the vertices of a square in the argand plane
To Prove: 2z2=1-iz1+1+iz3Proof:We need z2 in the form of z1 and z3
 So, we rotate about B
Rotation about B in clockwise direction:z1-z2=z3-z2e-iπ2⇒ z1-z2=z3-z2cosπ2-isinπ2⇒z1-z2=z3-z2-i⇒z1-z2=-iz3+iz2⇒z1+iz3=z2+iz2⇒z1+iz3=1+iz2⇒z11+i+iz31+i=z2⇒z11+i×1-i1-i+iz31+i×1-i1-i=z2⇒z11-i2+i+1z32=z2⇒2z2=z11-i+1+iz3Hence Proved
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