Consider Helly Theorem, taken from notes by Igor Pak:

Let $X_1, \dots, X_n \in {\mathbb{R}}^2$ be convex regions in the plane such that any triple interesects $X_i \cap X_j \cap X_k \neq 0$. Then there is a point in all the sets, $X_1 \cap \dots \cap X_n \neq \varnothing$.

This result is not obvious (although Pak's proof is short). However, any explicit collection of sets I build such that three of them intersect, have a clear total intersection. How about this simpler result, also from Pak's book:

Let $P_1, \dots, P_n \in {\mathbb R}^2$ be rectangles with sides parallel to the coordinate axes, such that any two intersect each other. Then all the rectangles have a nonempty intersection.

By Helly's theorem, we only need $n = 3$. What happens if we don't use Helly's theorem and try to prove this result directly?

Let $[x_1, x_1']\times [y_1, y_1'], \dots, [x_n, x_n']\times [y_n, y_n'] \in {\mathbb R}^2$ be rectangles in the plane, sides parallel to the $x,y$-axes, such that:

$x_i < x_j < x_i' < x_j'$ (or vice-versa) and $y_i < y_j < y_i' < y_j'$ (or vice-versa).

Then $x_i < x_j'$ for all $i,j$ and $y_i < y_j'$ for all $i,j$. So $[\mathrm{max} (x_i) , \mathrm{min} (x_i')] \times [\mathrm{max} (y_i) , \mathrm{min}( y_i')]$ is a rectangle that works.

Here, it wasn't hard to find that intersection point even without the reduction from Helly's theorem.

What kind of interesting collections of convex sets result in non-trivial uses of Helly's theorem?