Olympiad Euclidean Geometry

Suppose we have two convex plane figures $A$ and $B$, say in the Cartesian plane $\mathbb{R}^2$, for convenience (everything we do here will also apply to general convex sets). For example, imagine a circle (why convex?) and a square (or any convex polygon in general).

Then the common intersection $A\cap B$ of figures $A$ and $B$ is convex too! Why?

Let's just use the definition of convexity. The convexity of $A$ just says that the line segment connecting any two points in $A$ lies completely in $A$. Similarly, the convexity of $B$ just says that given any two points in $B$, the connecting line segment lies completely in $B$.

But now suppose we have two points in the intersection $A\cap B$. Convexity of $A$ says the connecting line segment lies completely in $A$; convexity of $B$ says it also lies completely in $B$. So it lies in both $A$ and $B$; in other words, by definition of intersection, it lies completely in the intersection $A\cap B$. So $A\cap B$ indeed satisfies the definition of convexity (i.e. $A\cap B$ is convex).