I've begun reading Stein's Complex Analysis book and found this in a proof of Goursat's theorem for a triangle. It mentions the 'diameter of a triangle' without explaining what is meant by that? Is it the diameter of the inscribed circle, or perhaps the superscribed circle? Or something else entirely? In case you're wondering, Figure 1 doesn't show this diameter.
The diameter of any nonempty subset of $\mathbb C$ (or $\mathbb R^n$, or of a metric space...) is $\operatorname{diam}A=\sup_{x,y\in A}|x-y|$, as njguliyev said. Opinions may vary on what the diameter of empty set should be.
It is interesting to compare $\operatorname{diam}A$ with the diameter of the smallest disk $C$ containing $A$, where $A$ is a planar set. Clearly, $$\operatorname{diam}A\le \operatorname{diam}C$$ with equality attained when $A$ contains a pair of opposite points of $C$. In the converse direction, the following is true (but is not straightforward to prove): $$\operatorname{diam}C\le \frac{2}{\sqrt{3}}\operatorname{diam}A$$ Equality holds when $A$ is an equilateral triangle.
It should be noted that when $A$ is a triangle, $C$ does not always coincide with the superscribed circle.
