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)sin² α/2

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|  e^i(∏-α)/2   ...........1

and  also (z2-z3)/|z2 -z3| = (z2-z1)/|z2-z1|  e^i(∏-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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