A3. Below you see a regular nonagon with all its diagonals.

How many isosceles triangles are there whose three different vertices are also vertices of the nonagon?

(A triangle is isosceles if two or three of its sides have the same length.)
A) 27
B) 30
C) 33
D) 36
E) 39

![](https://cdn.mathpix.com/cropped/2024_04_17_dd45dfb811a24abc1595g-1.jpg?height=348&width=368&top_left_y=1639&top_left_x=1529)
A3. B) 30

First, we count the number of equilateral triangles: there are 3 of them. Next, we count the isosceles triangles that are not equilateral. For such a triangle, there are 9 possibilities for the 'top'. Then there are 3 possibilities for the 'base'. This gives $9 \times 3=27$ possibilities. In total, there are thus $3+27=30$ isosceles triangles.

![](https://cdn.mathpix.com/cropped/2024_04_17_ecd98442595314acd6b1g-1.jpg?height=357&width=369&top_left_y=1232&top_left_x=1523)