The following figure shows a regular dodecagon.

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-14.jpg?height=384&width=383&top_left_y=1373&top_left_x=718)

Answer the following questions:

a) How many equilateral triangles can be formed such that their three vertices are vertices of the dodecagon?

b) How many scalene triangles can be formed such that their three vertices are vertices of the dodecagon?
Solution

We can inscribe the regular dodecagon in a circle as shown in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-44.jpg?height=394&width=391&top_left_y=2087&top_left_x=904)

Then the arcs corresponding to each side of the dodecagon must measure $(360 / 12)^{\circ}=30^{\circ}$.

a) The equilateral triangles we can form will also be inscribed in the circle. We know that the measure of the arc corresponding to a side of an equilateral triangle inscribed in a circle must be $(360 / 3)^{\circ}=120^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-45.jpg?height=421&width=473&top_left_y=662&top_left_x=673)

If we fix a vertex, say $A$ in the graph, there will then be a unique equilateral triangle using the vertex $A$.

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-45.jpg?height=442&width=391&top_left_y=1397&top_left_x=711)

We can thus count 12 triangles (one for each vertex). In this case, each triangle would be counted 3 times. We conclude that there are $12 / 3=4$ equilateral triangles inscribed in the dodecagon.

b) To count the number of scalene triangles, we can count the total number of triangles and subtract the number of isosceles and equilateral triangles.

First, we will count the isosceles triangles that are not equilateral. Fix a vertex, say, $A$. There are 4 isosceles triangles that are not equilateral inscribed in the dodecagon where vertex $A$ corresponds to the unequal angle.
![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-46.jpg?height=982&width=1044&top_left_y=229&top_left_x=586)

We can count, thus, 4 isosceles triangles for each vertex of the dodecagon, that is, $4 \times 12$ isosceles triangles that are not equilateral in total.

Now let's count the total number of triangles. To determine a triangle, we must choose three vertices from the set of 12 vertices of the dodecagon. For choosing the first vertex of the triangle, we have 12 options, for the second, 11, and for the third, 10 remain. But in the product $12 \times 11 \times 10$, we are counting the same triangle multiple times. Observe that there are six possible ways to order the vertices of a triangle $A B C$, which are listed below:

$$
A B C, \quad A C B, \quad B C A, \quad B A C, \quad C A B \quad \text { and } \quad C B A
$$

Therefore, in the product $12 \times 11 \times 10$, each triangle is being counted 6 times. We conclude that there are, in total, $\frac{12 \times 11 \times 10}{6}=220$ triangles inscribed in the dodecagon.

Thus, the number of scalene triangles will be equal to

$$
220-4 \times 12-4=168
$$

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