I am aware of this question, but mine is a bit more specific.
I want to find the coordinates of the Fermat point for a given triangle. Assuming that no angle in the triangle is larger than 120 degrees, there's an algorithm for doing that that goes something like this:
- Construct equilateral triangles on two sides of our given (original) triangle.
- Connect the new vertexes with the opposite vertexes from the original triangle.
- Find the point in which the connecting lines intersect - that's the Fermat point of our original triangle.
(paraphrased from Wikipedia)
Now, I want to do this programmatically, so I can't do it by drawing. It would be ideal if I had a formula for the coordinates of the Fermat point, but I've tried Google and I can't find it.
Anyway, what I plan on doing is to write a program that does something like this:
- Calculate the coordinates of the third vertex in the equilateral triangles for two sides of the original triangle.
- Calculate the equations of the lines that connect the new vertexes with the opposite vertexes from the original triangle. I can do this because two point uniquely determine a line.
- Given the two line equations that I got in step 2, calculate the coordinates of the intersection of those lines.
Am I missing something? The fact that I can't find a formula for this makes me think that I am, because, if what I'm trying to do is correct, why isn't there a direct formula? I mean, what I'm doing in each step is basically using a formula on what I got from the previous step, so why wouldn't there be a formula to go from step 0 (coordinates of the original triangle) to step 3 (coordinates of the Fermat point)?
Furthermore, if there isn't a formula and if I'm not missing anything, is there a simpler way of doing what I'm trying to do?