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 .

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mycroft holmes · Answer

Let A represent the vertex z 1 , with B and C similarly defined. Let z be the point P We can write z = t 1 z 1 +t 2 z 2 +t 3 z 3 = t 1 z 1 +t 2 z 2 + (1-t 1 -t 2 )z 3 so that z-z 3 = t 1 (z 1 -z 3 )+ t 2 (z 2 -z 3 ). Since t 1 ,t 2 >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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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,z₃are three pairwise distinct complex numbers and t₁,t₂,t₃are 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,z₃or on its boundary .

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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 .

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Let z₁,z₂,z₃are three pairwise distinct complex numbers and t₁,t₂,t₃are 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,z₃or on its boundary .