Question

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Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

oh, yes

Answer 5

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

Answer 6

acute triangles

Answer 7

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

Answer 8

we may use the relation between inscribed angle and central angle : \(\theta = \dfrac{\alpha}{2}\)

Answer 9

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

Answer 10

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.

Answer 11

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

Answer 12

thanks so much! i really appreciate it!

Answer 13

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.

Answer 14

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
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total 36 cases.