-3

How many triangles are there in this figure? And is there a formula? Figure

I found: ABC-ABD-ABG-AFG-ACD-ACG-AEG

BCF-BCG-BDG

CDG-CEG

That is, a total of 12. But not sure if I am missing some.

JMP
  • 21,771
Tobias Witte
  • 11
  • Looks like you missed four: ABE-ACF-BCE-BFG. – user14111 Oct 28 '22 at 10:11
  • 1
    There are $6$ lines, hence a maximum of $\binom63=20$ triangles. There are $4$ concurrencies (A,B,C,G) (where $3$ lines meet at a point), and as these cannot form triangles, there are $20-4=16$ triangles in total. Use the combinations of lines to identify them. – JMP Oct 28 '22 at 10:21
  • There are 6 small , 3 medium size and 6 large triangles and the triangle ABC. It sums to 16. – Etemon Oct 28 '22 at 12:58

1 Answers1

0

It takes $3$ Lines to form a triangle. You have $6$ lines in the figure, so if each combination of $3$ lines formed a triangle, you would have $\binom63=20$ triangles. Three lines failes to form a triangle if two of them don't meet, or if all three of them meet at one point. All of your lines meet, but you have three lines meeting at points $A$, $B$, $C$, and $G$, so the number of triangles in the figure is $\binom63-4=20-4=16$; they are the $12$ you listed plus the $4$ you missed, namely $ABE$, $ACF$, $BCE$, $BFG$.

user14111
  • 951