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Let z 1 ,z 2 ,z 3 are three pairwise distinct complex numbers and t 1 ,t 2 ,t 3 are non negative real numbers such that t 1 +t 2 +t 3 =1. prove that the complex number z=t 1 z 1 +t 2 z 2 +t 3 z 3 lies inside a triangle with vertices z 1 ,z 2 ,z 3 or on its boundary .

Let z1,z2,z3 are three pairwise distinct complex numbers and t1,t2,t3 are non negative real numbers such that t1+t2+t3=1. prove that the complex number z=t1z1+t2z2+t3z3 lies inside a triangle with vertices z1,z2,z3 or on its boundary .

Grade:12th pass

1 Answers

mycroft holmes
272 Points
7 years ago
Let A represent the vertex z1, with B and C similarly defined. Let z be the point P
 
We can write z = t1z1+t2z2+t3z3 =  t1z1+t2z2+ (1-t1-t2)z3 so that z-z3 = t1(z1-z3)+ t2(z2-z3).
 
Since t1,t2>0, this means the vector PC lies in the region between vector AC and BC.
 
Similarly, you can see that PA lies in the interior of BA and CA, and PB lies in the interior of AB and CB.
 
The intersection of all these regions in the triangle ABC and its interior thus proving the statement

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