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if A(Z 1),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)sin 2 α/2

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

Grade:11

1 Answers

Badiuddin askIITians.ismu Expert
148 Points
12 years ago

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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