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

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mycroft holmes · Answer

We can rewrite as z = t 1 (z 1 -z 3 ) + t 2 (z 2 -z 3 ). z 1 -z 3 and z 2 -z 3 are two of the sides of the triangle. Since t 1 and t 2 are positive, z will lie in the interior of the region described by the vectors z 1 -z 3 and z 2 -z 3 . In the same way one can see that z lies in the interior of the region formed vectors z 1 -z 2 and z 3 -z 2 as well as in the interior of that formed by z 2 -z 1 and z 3 -z 1 . Hence z lies in the interior of the triangle formed by z 1 , z 2 , and z 3 .

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

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

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

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

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