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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.

Arun
25763 Points
3 years ago
Dear Saurabh

We can rewrite as

z = t1 (z1-z3) + t2(z2-z3). z1-zand z2-z3 are two of the sides of the triangle. Since t1 and t2 are positive, z will lie in the interior of the region described by the vectors z1-zand z2-z3.

In the same way one can see that z lies in the interior of the region formed vectors z1-zand z3-z2 as well as in the interior of that formed by z2-zand z3-z1.

Hence z lies in the interior of the triangle formed by z1, z2, and z3.

Regards