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Let A ,B ,C ,D be four concyclic points ...

Let `A ,B ,C ,D` be four concyclic points in order in which `A D : A B=C D : C Bdot` If `A ,B ,C` are repreented by complex numbers `a ,b ,c` representively, find the complex number associated with point `Ddot`

Text Solution

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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)`
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