SOLUTION: Consider a regular n-gon, with all of its diagonals, where n > 4. A triangle in the n-gon consists of 3 vertices on the n-gon's perimeter and the 3 edges (from the perimeter of th
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Question 601005: Consider a regular n-gon, with all of its diagonals, where n > 4. A triangle in the n-gon consists of 3 vertices on the n-gon's perimeter and the 3 edges (from the perimeter of the n-gon or its diagonals) connecting these vertices to each other. How many triangles in the n-gon have more than one diagonal as edges?
You can put this solution on YOUR website! The total number of triangles we can form is . Here, we can subtract the triangles that have exactly one diagonal as an edge.* For this to occur, we need two edges of the triangle on the perimeter of the n-gon, which occurs if and only if all three vertices are consecutive. The number of such triangles is simply n (because we can fix a middle vertex, there are n vertices).
Therefore the number of triangles with more than one diagonal as an edge is
*We cannot have any triangles with all three edges on the perimeter.
