Z1,Z2, z3 are three pair wise distinct complex numbers and t1, t2, t3, are non-negative real numbers such that t1+ t 2 +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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Arun · Answer

Dear Saurabh 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 . Regards Arun (askIITians forum expert)

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Z1,Z2, z3 are three pair wise distinct complex numbers and t1, t2, t3, are non-negative real numbers such that t1+ t 2 +t3=1 Prove that the complex number z = t1z1 +t2z2 + t3z3 lies inside a triangle with vertices z1, Z2, z3 or on its boundary.

Z1,Z2, z3 are three pair wise distinct complex numbers and t1, t2, t3, are non-negative real numbers such that t1+ t 2 +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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Z1,Z2, z3 are three pair wise distinct complex numbers and t1, t2, t3, are non-negative real numbers such that t1+ t 2 +t3=1 Prove that the complex number z = t1z1 +t2z2 + t3z3 lies inside a triangle with vertices z1, Z2, z3 or on its boundary.

Z1,Z2, z3 are three pair wise distinct complex numbers and t1, t2, t3, are non-negative real numbers such that t1+ t 2 +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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