Question

State whether the following statement is true or false.

Let $z_1,z_2,z_3$ are three pair wise distinct complex numbers and $t_1,t_2,t_3$ are non-negative real numbers such that $t_1+t_2+t_3=1$. then 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.

• A. True
• B. False

Answer 1

The correct option is A True

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 $z_3$.