If you wnat a point that is equidistant from all three points in a triangle, then all you have to do is solve a series of equations:
D = SQR((X1-X4)^2 + (Y1-Y4)^2 + (Z1-Z4)^2)
D = SQR((X2-X4)^2 + (Y2-Y4)^2 + (Z2-Z4)^2)
D = SQR((X3-X4)^2 + (Y3-Y4)^2 + (Z3-Z4)^2)
1, 2, and 3 are the points of the triangle. 4 is the centerpoint.
Those equations calculate the distance between each corner point and the center point. We don't know the center point, but we do know that the distance is the same for all. D is the distance, and it is found in all three equations.
As we don't actually need to find the distance, we can simplify the equations somewhat right off the bat:
D = (X1-X4)^2 + (Y1-Y4)^2 + (Z1-Z4)^2
D = (X2-X4)^2 + (Y2-Y4)^2 + (Z2-Z4)^2
D = (X3-X4)^2 + (Y3-Y4)^2 + (Z3-Z4)^2
So now we're looking for D^2 for each set of points.
Unfortunately, I don't know how to proceed from here. Unlike myself and many folks here, the folks on #math on IRC are rude, elitist, and unhelpful and refused to answer my "too-easy for them" questions about how to simplify equations when you have stuff that is squared in it. So I don't know how to extract X4 Y4 and Z4 from the parentheses so I can get them on the other side of the equation.
What I did get out of them though is that sqr((x-z)^2) is not (x-z), it is |x-z|. So that cuts off one avenue I was considering for solving it.
D = SQR((X1-X4)^2 + (Y1-Y4)^2 + (Z1-Z4)^2)
D = SQR((X2-X4)^2 + (Y2-Y4)^2 + (Z2-Z4)^2)
D = SQR((X3-X4)^2 + (Y3-Y4)^2 + (Z3-Z4)^2)
1, 2, and 3 are the points of the triangle. 4 is the centerpoint.
Those equations calculate the distance between each corner point and the center point. We don't know the center point, but we do know that the distance is the same for all. D is the distance, and it is found in all three equations.
As we don't actually need to find the distance, we can simplify the equations somewhat right off the bat:
D = (X1-X4)^2 + (Y1-Y4)^2 + (Z1-Z4)^2
D = (X2-X4)^2 + (Y2-Y4)^2 + (Z2-Z4)^2
D = (X3-X4)^2 + (Y3-Y4)^2 + (Z3-Z4)^2
So now we're looking for D^2 for each set of points.
Unfortunately, I don't know how to proceed from here. Unlike myself and many folks here, the folks on #math on IRC are rude, elitist, and unhelpful and refused to answer my "too-easy for them" questions about how to simplify equations when you have stuff that is squared in it. So I don't know how to extract X4 Y4 and Z4 from the parentheses so I can get them on the other side of the equation.
What I did get out of them though is that sqr((x-z)^2) is not (x-z), it is |x-z|. So that cuts off one avenue I was considering for solving it.