# given that e^iA,e^iB,e^iC, are in a.p ,where A,B,C are the angles of the triangle then the triangle is:1)isoceles2)equilateral      ORright angled

Aman Bansal
592 Points
11 years ago

Dear Pragya,

We have that
2exp(iB)=exp(iA)+exp(iC)
or
-2exp(-i(A+C))=exp(iA)+exp(iC)
Hence
xy(x+y)=-2
where x=exp(iA) and y=exp(iB)
taking mod on both sides
|x+y|=2 ...since |x|=|y|=1
But we have the inequality
|x+y|<=|x|+|y|=2

the maxima is achieved when x and y are in phase.Hence
x=k.y where k is a positive real
But |x|=|y|=1 Hence x=y
or
exp(iA)=exp(iC)
or A=C
We also have
2exp(iB)=exp(iA)+exp(iC)=2exp(iA)
which gives A=B.
Hence A=B=C..equilateral

Best Of luck

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Thanks

Aman Bansal