proving there exists no triangle with 3 vertices in specific area in a 3 x 3 triangle Can you take a look at this question?:
"Suppose figure shown below. An equilateral triangle is divided into 9 smaller equilateral triangles. Prove that there exists no equilateral triangle with all vertices in the interior of the orange area, but not totally inside just one of them ( vertices must be inside at least 2 orange regions)."

 A: I enumerated the small triangles so that it would be easier to explain my approach. Also, let $A,B,C$ be the vertices of the triangle that we will construct by putting them inside orange triangles. WLOG, also assume small sides have length $1$ for convenience of the argument about to be given.

Case 1 ($A,B,C$ are inside distinct orange triangles): Since there is a symmetry and we don't have many cases to consider here, WLOG first suppose $A$ is inside $2$ and $B$ is inside $4$. Then since $C$ should be inside either $5$ or $9$, the triangle we construct this way has wide angle, therefore cannot be equilateral.
Now, suppose $A$ is inside $2$ and $B$ is inside $5$. Now, say $|AB| = x$, where $0 < x < 2$ and notice that in order to construct a equilateral triangle, $C$ must be inside $6$ or $7$ (when we take $x = 2$, $C$ is on the point where $7$ and $9$ intersect each other but $x < 2$ so it cannot go further than the triangle $7$. Otherwise, if $x$ is small, $C$ will be inside $6$), which are not orange. Therefore we cannot construct an equilateral triangle.
Case 2 ($A,B$ are inside the same orange triangle, $C$ is inside distinct one): First, WLOG, suppose $A,B$ are inside $2$. Then suppose $|AB| = x$ but this time, we have $0 < x < 1$. In this case we must have $|AC| = |BC| < 1$ obviously. But this constraint tells us that $C$ cannot go further than adjacent triangles of $2$, which are $3$ and $6$. Also notice that there is no two orange triangles that are adjacent. So $C$ will either be inside $2$ or $C$ will be inside $3$ or $6$. But both of these are not allowed (first one is not allowed because $A,B,C$ are inside the same orange triangle, second one is not allowed because $C$ is inside white triangle) so we cannot construct an equilateral triangle in this case.
Since there is no other case to consider, we cannot construct an equilateral triangle with given constraints.
