0
votes
1answer
43 views

Number of triangles formed by all chords between $n$ points on a circle

We have $n$ point on circumference of a circle. We draw all chords between this points. No three chords are concurrent. How many triangles exist that their apexes could be on circumference of ...
4
votes
1answer
129 views

Proof of “Japanese Theorem” — Triangulation of Cyclic Polygon

On Mathoverflow, I saw this great result on the "Japanese Theorem". “Japanese Theorem” on cyclic polygons: Higher-dimensional generalizations? Given triangulation of a cyclic polygon, the sum of ...
1
vote
2answers
48 views

Number of Equilateral triangles in circle with 42 evenly spaced points?

I know that the answer is 42/3 = 14 points, or in general for a circle with N points it is N/3, but I don't know why it actually works. Why is the number of equilateral triangles for a circle with N ...
1
vote
1answer
50 views

Number of ways to form isosceles triangle by picking points on a circle

Given a circle with 24 evenly spaced points, how would you find the number of possible isosceles triangles (which includes equilateral) that can by drawn using the points? My attempt was to say that ...
1
vote
2answers
41 views

Pidgeonhole principle - I'd like an explanation for this answer

A friend of mine showed me how to solve this question: suppose there are 5 black dots drawn on a blue sphere. show that there is a closed hemisphere such that 4 of the black points are in it. his ...
0
votes
1answer
2k views

How many triangles can be formed from N points on a circle?

I have a circle with N points on it, and I want to determine how many triangles can be formed using these points. How can I do this? Thanks! Andrew
41
votes
4answers
3k views

Do circles divide the plane into more regions than lines?

In this post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. What happens if we use circles instead of lines? That is, what ...
2
votes
2answers
2k views

Counting number of distinct regions with intersecting circles

Given $n$ circles of possibly different radii, how many distinct regions can there be? For small $n$, I can work it out with pictures. (I'm pretty sure $n=4$ can yield 13 distinct regions, but not ...