Let z=a+bi,w=c+di,a,b,c,d∈R.
Then
∣z+w∣2=(a+c)2+(b+d)2
∣z−w∣2=(a−c)2+(b−d)2
∣z+w∣2−∣z−w∣2=(a+c)2+(b+d)2
−((a−c)2+(b−d)2)=4ac+4bd
=4(ac+bd)
zw^=(a+bi)(c−di)=ac+bd+(−ad+bc)i
Re(zw^)=ac+bd Hence
∣z+w∣2−∣z−w∣2=4(ac+bd)=4Re(zw^),
a,b,c,d∈R Therefore
∣z+w∣2−∣z−w∣2=4Re(zw^)
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