Let's tackle this problem step by step to show why a player who adds the 18th edge wins.

Step 1: Analyze the Diagram

Consider the triangle with 3 vertices and 6 additional vertices inside it. The total number of vertices is $9$. The players will be drawing edges between these vertices.

Step 2: Maximum Number of Edges Without Crossing

To understand the game, it's important to know the maximum number of edges that can be drawn without violating the rules (no crossing edges, no repeated edges between the same pair of vertices).

For any $n$ points in general position (i.e., no three points are collinear), the maximum number of non-crossing edges that can be drawn is given by:

$\text{Maximum number of non-crossing edges} = 3n - 6$

In our case, $n = 9$ (since we have 9 vertices), so the maximum number of non-crossing edges is:

$3(9) - 6 = 27 - 6 = 21$

This calculation gives us the upper bound, but remember that some edges are already used up by the original triangle.

Step 3: Pre-Existing Edges

The problem states that the 3 vertices of the triangle and its 3 edges are already drawn. So we already have 3 edges.

Step 4: Number of Edges Left to Draw

Given that the maximum number of non-crossing edges is 21 and 3 edges already exist, the players can draw a maximum of:

$21 - 3 = 18 \text{ new edges}$

Step 5: Win Condition

The players alternate turns to draw edges. If a player draws the 18th edge, no more edges can be added according to the rules. Thus, the other player has no legal moves left and loses the game. Therefore, the player who adds the 18th edge wins.

Conclusion

The player who adds the 18th edge wins because this move leaves the opponent without any legal moves to make.

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Would you like to explore more details about the solution, or do you have any questions about the logic used?

Here are some additional questions to expand on this topic:

1. How would the strategy change if we had a different number of interior vertices?
2. What if the players could only add one edge per turn?
3. What happens if three vertices are collinear inside the triangle?
4. How does the strategy differ if the initial triangle had more than three edges?
5. Can you generalize this strategy to any polygon?

Tip: In combinatorial games like this, always calculate the maximum number of possible moves to understand the win/loss conditions clearly.