Let's say we have three exact locations on the Earth. Let's say people from those three locations want to meet in a point that is between all three locations, but also the same distance from each location.
Center of mass/triangle center doesn't work because when one location is much further than the other two, the following triangle is produced:
Obviously as seen in the picture, the vertex on the left is much further away from the mass center than those on the right.
What equation, given coordinates $(x,y)$ of each vertex, could give me a point that would be the same distance from each vertex? If it has to, the point could be also outside the triangle, just I want it to be equal distance from each vertex.
Best Answer
The point you want is called the circumcenter, it is the intersection of all three perpendicular bisectors of the sides of the triangle. The other common important points of triangles are probably:
center of mass: point of intersection of the three medians (always inside triangle)
orthocenter: point of intersection of the three heights
incenter: point of intersection of the three angle bisectors (always inside triangle)
It can be proven that the circumcenter, center of mass and orthocenter lie on the same line called Euler's line.