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
.