The three triangles in the figure intersect at 12 different points. What is the maximum number of intersection points of any three triangles?

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-023.jpg?height=334&width=609&top_left_y=1289&top_left_x=832)
When we add a new triangle to a figure composed of triangles, it intersects each of the sides of the existing triangles at, at most, two points. Initially, starting with a single triangle, we have no intersection points. By adding a second triangle, we introduce, at most, \(2 \times 3 = 6\) intersection points. Similarly, by adding a third triangle, we introduce, at most, \(2 \times 6 = 12\) more intersection points. Therefore, three triangles can intersect at, at most, \(6 + 12 = 18\) points. The figure shows that this case of 18 intersection points can indeed occur.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-083.jpg?height=434&width=442&top_left_y=2262&top_left_x=887)