# Smallest triangle in a convex polygon triangulation

I have been working on this problem for quite a while and it seems necessary to prove or disprove this particular problem.

Suppose $T$ is the set of all possible triangles made from the vertices of a given convex polygon $A$.

If $A$ has 3 to 4 vertices, the smallest triangle in $T$ will always be made from three adjacent vertices.

Is true for all convex polygons? If not, what can we state about the minimum triangle?

Suppose that you have a triangle where not all 3 vertices are adjacent. Let $v$ be a vertex that is not adjacent to either of the other two. Let $a,b$ be the other two vertices. Then the area of the triangle is one half the length of $(a,b)$ multiplied by the height from $(a,b)$ out to $v$. But if $v$ is replaced with one of $v$'s neighbors, then one of the two neighbors must produce a lower height and hence a lower area triangle. Thus all 3 vertices must be adjacent in a minimum area triangle.