Convexity of the 2-Dimensional disc

Question

How can I prove the convexity of the disk $ |z-z_0| \leq R$ algebraically?

I know I have to prove that a linear expression of two points in the disk is still in the disk.

Answer 1

If $\bigl\lvert z_1-z_0\bigr\rvert\leqslant R$ and $\bigl\lvert z_2-z_0\bigr\rvert\leqslant R$ , then, for every $\lambda\in[0,1]$,\begin{align}\bigl\lvert\lambda z_1+(1-\lambda)z_2-z_0\rvert&=\lvert\lambda z_1-\lambda z_0+(1-\lambda)z_2-(1-\lambda)z_0\bigr\rvert\\&\leqslant\lambda\bigl\lvert z_1-z_0\bigr\rvert+(1-\lambda)\bigl\lvert z_2-z_0\bigr\rvert\\&\leqslant\lambda R+(1-\lambda)R\\&=R.\end{align}