# How many triangles in picture?

Can you tell me how many triangles are in this picture? I've counted 96, not sure that I got right answer.

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Perhaps you could try counting them by using the symmetry of the shape (if you haven't already). – Shaun Nov 21 '13 at 13:50
Clearly there are 19 unique points. Number of triangles that can be formed from 19 points taking 3 at a time=$19\choose3$. Also note that some of the points lie on a straight line. When we took $19\choose3$ we included the ones that lie on a straight line. Perhaps finding the number of lines and subtracting that from $19\choose3$ will yield the answer. – GTX OC Nov 21 '13 at 14:05
@GTXOC I see 19 unique points... – apnorton Nov 21 '13 at 14:08
Alright I missed the ones in the middle of the base. – GTX OC Nov 21 '13 at 14:09
@All: Not all triangles are valid even if the points could form one, consider lines between "star-corners". – AlexR Nov 21 '13 at 14:11

Let's see! We'll break it up into the following cases (where an apex means one of the six points of the star):

A: Triangles that contain three apexes
There are 2 of these.

B: Triangles that contain a pair of opposite apexes
For each such pair, there are 2 of these, so 6 in total.

C: Triangles that contain a pair of non-opposite apexes
For each non-adjacent apex pair, there are three of these, so 18 in total.

D: Triangles that contain exactly one apex
For each apex, I count 10 such triangles, so 60 in total.

E: Triangles that contain no apex
For each edge of the internal hexagon, there are 3 triangles, so 18 in total.

I make that 2 + 6 + 18 + 60 + 18 = 104.

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I verified and this seems to be correct. – AlexR Nov 21 '13 at 14:16
Look OK to me too, though categories $D$ and $E$ could be somewhat refined. – Marc van Leeuwen Nov 21 '13 at 14:24
@Marc: I agree with you about category D. – TonyK Nov 21 '13 at 14:29
A small reality check: If you dismiss the two big equilateral triangles, all other triangles belong to orbits of $6$ under rotations by $2\pi/6$, so $T-2$ must be divisible by $6$, which $T=104$ is but $T=96$ (the OP's original count) is not. – Barry Cipra Nov 21 '13 at 14:54
For Category $D$, you can take the apex at the very top and count triangles that are and aren't symmetric across the "$y$"-axis. There are $2$ that are and $4$ pair that aren't. – Barry Cipra Nov 21 '13 at 15:13

I only get 98. I am missing the 3rd triangle under section C. I only see 2 triangles that contain a pair of non-opposite apexes.

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