Question

Let z1 , z2 , z3 are three pair wise distinct complex numbers and t1 , t2 , t3 are non-negative real numbers such that t1 + t2 + t3 = 1. Prove that the complex number z = t1 z1 + t2 z2 + t3 z3 lies inside a triangle with vertices z1 , z2 , z3 or on its boundry.

Answer 1

Answer:

Step-by-step explanation:

Given Let z1 , z2 , z3 are three pair wise distinct complex numbers and t1 , t2 , t3 are non-negative real numbers such that t1 + t2 + t3 = 1. Prove that the complex number z = t1 z1 + t2 z2 + t3 z3 lies inside a triangle with vertices z1 , z2 , z3 or on its boundry.

 The to sides of a triangle are Z1 - Z3 and Z2 - Z3, Z lies in the interior of region described by vectors Z1 - Z3 and Z2 - Z3, since t1 and t2 are positive.

So similarly Z lies in the interior of region formed by the vectors Z1 - Z2 and Z3 - Z2 as well as interior of that formed by Z2 - Z1 and Z3 - Z1

Therefore Z lies in the interior of the triangle formed by Z1 , Z2 and Z3