0
$\begingroup$

I have three circles. One is at $(0,0)$ and has radius $n$, another has is at $(1,0)$ and has a radius $m$, and the third is at $(0.5, \sqrt{0.75}))$ and has a radius of $o$. All of the radius values are integers.

How can I compute where these three circles intersect, for any value of $n$, $m$, and $o$? Edit: The values here are outputs, not inputs- that is, I want to determine possible values for $n$, $m$, $o$, not give them. For example, there are obviously no solutions with $n = m = o = 1$.

Bonus points for any solution that can solve for more than three circles under the same constraints.

Sorry: I want to know how to find the point(s) at which the three circles meet, if there are any.

Edit: My mistake! The initial positions for the circles actually violate the original constraint. The second circle is positioned at $(1, 0)$.

I appear to have asked a question of significantly higher difficulty than I had first imagined. My apologies.

Edit:

$$x^2 + y^2 = n^2$$ $$x^2 + y^2 - 2x + 1 = m^2$$ $$1 - x - \sqrt3y + y^2 + x^2= o^2$$

Therefore,

$$-n^2 = 1 - 2x - m^2 = 1 - \sqrt3y - x - o^2 = -x^2 - y^2$$ $$ n^2 = m^2 + 2x - 1 = o^2 + \sqrt3y + x = x^2 + y^2$$

This places a number of simultaneous conditions if $m$, $n$, and $o$ are to be integral. I don't see how any x and y could satisfy this equation for any integral m, n, o.

  • 1
    Are you looking for (a) the point where all three of them intersect if it exists? (b) the (up to 6) points where pairs of them intersect? (c) the area where all three overlap?2012-02-17
  • 0
    @Henry: a. I guess I'll try to make the question clearer.2012-02-17

2 Answers 2