1. ## How many triangles?

Hi,

How many triangles are in this picture:

Kind regards,

Kepler

2. Originally Posted by kepler
Hi,

How many triangles are in this picture:

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Kind regards,

Kepler
Nice problem. I might have counted wrong.

Spoiler:

There are 9 vertices/points of intersection, and consider sets of three of them. We have 3+1+2+2*C(4,3) = 14 sets that are collinear, where C(n,k) is binomial coefficient. Then consider sets of two of them. I count 6 that are not connected. Each of these 6 would have given 7 triangles, had they been connected. So I get C(9,3) - 14 - 6*7 = 28 total.

3. Hi,

yep, that's my guess too.

Kind regards,

Kepler

4. Originally Posted by kepler
Hi,

How many triangles are in this picture:

Kind regards,

Kepler
Counting sequentially......

If there are no internal lines, 1 triangle.
Draw the vertical line, 2 new triangles.
Draw the line from the left vertex to the right side, 5 new triangles.
Draw the line from the right vertex to the left side, 6 new triangles.
Draw the line from the left side to the mid-base, 7 new triangles.
Draw the line from the right side to the mid-base, 7 new triangles.

Total 28.

5. I found 34 triangles
picture of them:

6. Well done!

I miscounted.
My totals should have been....

No internal lines.....1 triangle
1 internal line........2 more
3 lines..................5 more
4 lines..................8 more
5 lines..................8 more
6 lines..................10 more

7. Originally Posted by beeliz
I found 34 triangles
picture of them:
33 and 34 are duplicating your 4 and 5.

I agree with the total, though...

8. Here's another counting process

9. Originally Posted by tom@ballooncalculus
33 and 34 are duplicating your 4 and 5.

I agree with the total, though...
whoooops. well I found 34 somewhere...

oh in the image Archie Meade posted, it's #2 in the fourth triangle and #1 in the sixth triangle that I missed in my picture. but I found them the first time I counted during a boring english class

better picture: because I'm a perfectionist