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`        if A(Z1),B(Z2),C(Z3) on complex plane such that angles at B and C are each equal to 1/2(∏-α) then prove that (Z2-Z3)2 =4(Z3 - Z1)(Z1-Z2)sin2 α/2`
7 years ago

147 Points
```										Dear deepika

<B =<c =(∏-α)/2    so angel A = α
now use sine rule in trinagle ABC
|z2 -z3|/sinα  = |z1-z3|/cosα = |z1-z2|/cosα
now by the rotation property of vector
(z2-z3)/|z2  -z3|  = (z1-z3)/|z1-z3|  ei(∏-α)/2   ...........1

and  also (z2-z3)/|z2  -z3| = (z2-z1)/|z2-z1|   ei(∏-α)/2   ...........2

multiple first and second equation
(Z2-Z3)2  /|z2  -z3|2=(Z3 -  Z1)(Z1-Z2)/|z1-z3||z2-z1|

now put the value of |z1-z3|, |z2-z1| in terms of |z2  -z3|
(Z2-Z3)2  =sin2α(Z3 -  Z1)(Z1-Z2)/cos2α/2
(Z2-Z3)2 =4sin2α/2 (Z3 -   Z1)(Z1-Z2)
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7 years ago
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