Let A,B, C,D be four concyclic points in order in which AD:AB=CD: CB. If A,B,C are representing by complex numbers a,b,c respectively find the complex number associated with point D.

Explanation

Let complex number representing point 'D' is d and `angleDAB = theta`.
So, ` angle BCD = pi - theta` (A,B,C,D are concyclie ),Now applying rotation formula on A and C, we get
`(b-a)/(d-a) = (AB)/(AD)e^(itheta)`
and ` (d-c)/(d-c) = (CD)/(CB) e^(i (pi-theta))`
Multiplying these two, we get
`((b-a)/(d-a)) ((d-c)/(b-c)) = (AB xx CD)/(AD xx CB)e^(ipi)`
`(d(b-a) - c(b-a))/(d(b-c)-a(b-c)) =-1" "(because (AD)/(AB) = (CD)/(CB))`
`or d = (2ac - b(a+c))/(a+c-2ab)`