title: circles and triangles problem
body: <p>I have an interesting problem here I've been trying to solve for the last little while:</p>
  
  <p>I have 3 circles on a 2D xy plane, each with the same known radius. I know the coordinates of each of the three centers (they are arbitrary and can be anywhere). </p>
  
  <p>What is the largest triangle that can be drawn such that each vertex of the triangle sits on a separate circle, what are the coordinates of those verticies?</p>
  
  <p>I've been looking at this problem for hours and asked a bunch of people but so far only one person has been able to suggest a plausible solution (though I have no way of proving it).</p>
  
  <p>The solution that we have come up with involves first creating a triangle about the three circle centers. Next we look at each circle individually and calculate the equation of a line that passes through the circle's center and is perpendicular to the opposite edge. We then calculate two intersection points of the circle. This is then done for the next two circles with a result of 6 points. We iterate over the 8 possible 3 point triangles that these 6 points create (the restriction is that each point of the big triangle must be on a separate circle) and find the maximum size.</p>
  
  <p>The results look reasonable (at least when drawn out on paper) and it passes the special case of when the centers of the circles all fall on a straight line (gives a known largest triangle). Unfortunate i have no way of proving this is correct or not.</p>
  
  <p>I'm wondering if anyone has encountered a problem similar to this and if so, how did you solve it?</p>
  
  <p>Note: I understand that this is mostly a math question and not programming, however it is going to be implemented in code and it must be optimized to run very fast and efficient. In fact, I already have the above solution in code and tested to be working, if you would like to take a look, please let me know, i chose not to post it because its all in vector form and pretty much impossible to figure out exactly what is going on (because it's been condensed to be more efficient).</p>
  
  <p>Lastly, yes this is for school work, though it is NOT a homework question/assignment/project. It's part of my graduate thesis (abet a very very small part, but still technically is part of it).</p>
  
  <p>Thanks for your help.</p>
  
  <p>Edit: Heres a new algorithm that i came up with a little while ago.</p>
  
  <p>Starting at a circle's centre, draw a line to the other two centres. Calculate the line that bisects the angle created and calculate the intersections between the circle and the line that passes through the centre of your circle. You will get 2 results. Repeat this for the other two circles to get a total of 6 points. Iterate over these 6 points and get 8 possible solutions. Find the maximum of the 8 solutions.</p>
  
  <p>This algorithm will deal with the collinear case if you draw your lines in one "direction" about the three points. </p>
  
  <p>From the few random trials i have attempted using CAD software to figure out the geometries for me, this method seems to outperform all other methods previously stated However, it has already been proven to not be an optimal solution by one of Victor's counter examples.</p>
  
  <p>I'll code this up tomorrow, for some reason I've lost remote access to my university computer and most things are on it.</p>