TITLE: How many triangles are there?
QUESTION [33 upvotes]: The question is how many triangles are there in the following picture?

I have thought to solve it by creating a formula based on the angles of the lines starting from the bottom of each side. I don't get it right though. Any clues/ideas would be appreciated.

REPLY [5 votes]: Using Burnside's lemma, the number of triangles is equal to $6n-22$ where $n$ is the number of orbits of the action of $D_3$ on the set of triangles. 
To count them, notice that : 

Every orbit contains one the four triangles in the bottom right-hand corner of the big triangle. 
Two triangles in the same orbit are isometric.

So you only need to count twenty-one orbits rather than one hundred and four triangles.