Clarissa drew the hexagon below and marked its vertices with the letters $A, B$, $C, D, E$ and $F$.

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-12.jpg?height=443&width=486&top_left_y=504&top_left_x=685)

Then she declared that some points of the hexagon would be called cool. A point of Clarissa's hexagon is called cool if it satisfies any of the properties below:

- this point is one of the vertices of the hexagon;
- this point is the intersection of two diagonals of the hexagon;
- this point is the intersection of any line segment connecting two other cool points.

For example, in the figure below, the points $A, B, D$ and $F$ are cool points, as they are all vertices of Clarissa's hexagon. The point $H$ is also a cool point because it is the intersection of the diagonal $\overline{A B}$ with the diagonal $\overline{B F}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-12.jpg?height=448&width=486&top_left_y=1578&top_left_x=685)

Clarissa says that a triangle contained in her hexagon is cool if all of its vertices are cool points. For example, in the figure below, Clarissa's hexagon is divided into four cool triangles.

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-12.jpg?height=434&width=486&top_left_y=2262&top_left_x=685)
a) Clarissa wants to divide a hexagon into six cool triangles. Show how she can do it.

b) Explain why the point I in the figure below is a cool point.

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-13.jpg?height=437&width=488&top_left_y=455&top_left_x=887)

c) Show a way in which Clarissa can divide the hexagon into 10 cool triangles.

d) Show a way in which Clarissa can divide the hexagon into 2014 cool triangles.
Solution

a) Consider the figure below that appears in the problem statement:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-37.jpg?height=437&width=486&top_left_y=1980&top_left_x=885)

In this figure, there are four legal triangles. If we add the diagonal $\overline{B D}$, it will create one more legal point $G$, and each of the triangles $F B C$ and $F C D$ will be divided into two legal triangles each. Thus, a total of six legal triangles will be obtained, as shown in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-38.jpg?height=448&width=505&top_left_y=227&top_left_x=684)

Note: There are other solutions!

b) The points $D, E$, and $F$ are legal because they are vertices of Clarissa's hexagon. Point $H$ is also legal, as it is the intersection of the diagonal $\overline{A D}$ with the diagonal $\overline{B F}$. Point $I$ is the intersection of the line segment $H E$ with the line segment $D F$. Since these segments connect legal points, point $I$ is also a legal point.

c) One possible way is, starting from the figure below, to connect point $H$ to point $C$ and point $B$ to point $D$, as shown below:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-38.jpg?height=440&width=488&top_left_y=1322&top_left_x=687)

thus obtaining the following figure in which the hexagon is divided into 10 legal triangles:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-38.jpg?height=442&width=488&top_left_y=2052&top_left_x=687)

A second solution would be to draw the diagonals $\overline{A D}, \overline{A E}, \overline{B D}, \overline{B E}, \overline{C F}$, resulting in the division of the hexagon into the 10 legal triangles drawn below:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-39.jpg?height=446&width=486&top_left_y=231&top_left_x=885)

Note: Other solutions are possible!

d) We illustrate here one possible way Clarissa can proceed. Other solutions are possible.

Clarissa should start by drawing the diagonals $\overline{A C}, \overline{B D}, \overline{D F}, \overline{E A}$, and $\overline{F B}$ in her hexagon. This will result in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-39.jpg?height=436&width=480&top_left_y=1027&top_left_x=888)

In the figure above, Clarissa's hexagon is divided into 12 legal triangles and one more hexagon with vertices $N, O, P, Q, S$, all of which are legal vertices.

Clarissa should now repeat the same procedure on the new hexagon, thus obtaining 12 more legal triangles and an even smaller hexagon whose vertices are all legal.

We conclude that each time Clarissa repeats this procedure, a total of 12 legal triangles will be added to the figure, and there will still be a hexagon whose vertices are legal points (always contained within the hexagon obtained in the previous step). After 167 repetitions, Clarissa will have divided her hexagon into $167 \times 12 = 2004$ legal triangles and will still have one remaining hexagon. She should then divide this remaining hexagon into 10 legal triangles using, for example, the procedure obtained in part (b). At the end of this entire process, she will have divided her hexagon into $2004 + 10 = 2014$ legal triangles.