Consider the circle |z|=r in the Argand plane, which is in fact the incircle of trinagle ABC. If contact points opposite to the vertices A,B,C are `A_(1)(z_(1)), B_(1)(z_(2))` and `C_(1)(z_(3))`, obtain the complex numbers associate with the vertices A,B,C in terms of `z_(1),z_(2)` and `z_(3)`.

Explanation

Let `A(z_(a)),B(z_(b))` and `C(z_(c))` be the vertices of the trinagle.

Considering the rotation about points `A_(1)` and `B_(1)`, respectively, we get
`(z_(c)-z_(1))/(0-z_(1)) =(|z_(c) - z_(1)|)/(r) e^(-pi//2) and (z_(c) - z_(2))/(0-z_(2)) = (|z_(c) - z_(2)|)/(r)e^(ipi//2)`
`((z_(c)-z_(1))z_(2))/(z_(1)(z_(c)-z_(2)))=e^(-ipi) = -1`
`rArr (z_(c)- z_(1))z_(2) = - z_(1)(z_(c)-z_(2))`
or `z_(c)= (2z_(1)z_(2))/((z_(1) +z_(2))`
Similarly,
`z_(a)=(2z_(2)z_(3))/((z_(2)+z_(3))` and `z_(b)=(2z_(1)z_(3))/((z_(1)+z_(3))`