## anonymous one year ago ***

1. ganeshie8

|dw:1439559062484:dw|

2. anonymous

not all of them will pass through the center though...

3. ganeshie8

Notice that the center of circle is the "circumcenter" of the triangle we're working on

4. ganeshie8

so basically the problem is equivalent to finding the triangles such that the circumcenter lies interior to the triangle

5. anonymous

oh, yes

6. ganeshie8

for what triangles do we have circumcenter interior to the triangle ?

7. anonymous

acute triangles

8. ganeshie8

yes, they would be all "acute" triangles so our job is to find the number of acute triangles and divide them by 84

9. ganeshie8

we may use the relation between inscribed angle and central angle : $$\theta = \dfrac{\alpha}{2}$$ |dw:1439560115696:dw|

10. ganeshie8

not really sure how to approach this, im still thinking..

11. anonymous

Thanks so much for helping me, sir. But I must go to sleep now, its getting late for me. I will think about this tomorrow.

12. ganeshie8

Okay, have good sleep :) I'll try to post the solution over the night, pretty sure this is not that hard...

13. anonymous

thanks so much! i really appreciate it!

14. Loser66

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15. Loser66

Consider vertex A. From AB, we have 7 triangles, and among them just ABF has center inside of it. That is the probability to get the center inside of the triangle is 1/7 for vertex A Same for others Hence the total is 1/7^9 but we have to subtract the overlap parts. I meant $$\triangle ABC$$ when consider node A will be overlap with $$\triangle BCA$$ for node B.

16. Loser66

Hence for node B, we have 1 triangle overlaps with node A for node C, we have 1triangles overlaps with node A $$\triangle ACD$$, and 1 triangle overlaps with nod B $$\triangle CBA$$ Same for other nodes and same argument, we have the logic 1st node --0 overlap 2nd node--1 overlap 3rd node---2 overlap ::::::::::::::::::::::::::::::: 9th node---8 overlap ------------------------ total 36 cases.

17. Loser66

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18. Loser66

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19. Loser66

Ok, part 1) is done, now just calculate the possibility of it.

20. ganeshie8

for part a, im getting $$\dfrac{30}{84}$$ for partb, the general formula for probability is $$\dfrac{n+1}{2(2n-1)}$$

21. anonymous

can you tell me how, i need to write an explanation with my answer...