23.3. Counting Diagonals in Convex Polygons

The third question is how many diagonals are there in a convex polygon with n sides? So, again we can derive the formula using induction or some other mechanism but we will count it directly.

This chunk introduces the problem of counting diagonals in a convex polygon with n sides. The goal is to find a formula or method for determining how many diagonals exist in such a polygon. The section suggests that a direct counting approach will be used instead of induction.

Imagine a simple triangle (with 3 sides) drawn on a piece of paper. In this case, there are no diagonals, as the only lines are the edges of the triangle itself. As we increase the number of sides (like to a square or pentagon), we start to see diagonals, or lines connecting non-adjacent vertices.

Now, let us focus on some arbitrary vertex v and try to count the number of diagonals that we can have where v is one of the end points. Now if v is one of the end points of the diagonal then the other end point of the diagonal cannot be the immediate neighbors of v...

By selecting one vertex of the polygon (let's call it v), the section explains the restrictions on choosing the other endpoint of the diagonal. The other endpoint cannot be one of the immediate neighbors of v, which limits the options available for drawing diagonals from that vertex. For a polygon with n sides, since v has two neighbors, it effectively leaves n - 3 vertices available as potential endpoints for the diagonals.

Imagine standing on a corner of a rectangular table (which represents a 4-sided polygon). You can reach out to touch the other corners, but can't touch the corners immediately next to you, because those are just the edges of the table. So, as you count possible connections, you realize you can only connect to the far corners.

So, that means with v as one of the end-points of the diagonal I can have n - 3 diagonals, why n - 3? Because excluding these 3 vertices namely v, v and v all the remaining n - 3 end points can be the other end point of the diagonal with v being one of the end points. So, now it turns out that the total number of diagonals will be (n(n - 3))/2.

This chunk calculates the total number of diagonals in the polygon. Since each vertex can be connected to n - 3 other vertices by a diagonal, and this is true for each of the n vertices, we start by multiplying n by (n - 3). However, this counts each diagonal twice (once from each endpoint), thus we divide by 2 to get the correct total number of unique diagonals, which leads us to the formula (n(n - 3))/2.

Think of the vertices as people standing in a circle. If every person tried to shake hands with everyone else (but not with their neighbors), they would end up shaking hands with most, but not all. By counting each handshake twice (once for each participant), you realize the total number of unique handshakes is less than just multiplying participants by available partners.

But I will be counting the diagonals twice; I will be counting some of the diagonals twice namely the diagonal with the end points v, v will be counted twice because with v being one of the end points...

This chunk emphasizes the importance of accounting for double counting in the diagonal calculation. Since each diagonal between two endpoints is counted from both directions (from vertex i to vertex j and vice versa), the section explains why we must divide the total by 2 to get the accurate count of diagonals.

Consider you and a friend each writing down everyone you've met at a party. If you both list each other, you have duplicate entries. To solve this, you decide only one of you writes down the attendees, giving an accurate count of the unique people who were there.

so this will be the case where n is greater than equal to 4 because if the number of sides is 3 or less than 3 then we cannot have any diagonal...

The section concludes by specifying that the formula and reasoning only apply when n is 4 or higher since polygons with 3 sides (triangles) do not have any diagonals. This defines the context in which we can use the previously derived formula to calculate the number of diagonals.

You can't connect the corners of a triangle with diagonals because the only lines are the edges themselves. It's only when you have at least a quadrilateral that you start to find new connections (or diagonals) formed by lines that aren't just the sides of the shape.

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