function center = triangle_circumcenter_2 ( t )
%*****************************************************************************80
%
%% triangle_circumcenter_2() computes the circumcenter of a triangle.
%
% Discussion:
%
% The circumcenter of a triangle is the center of the circumcircle, the
% circle that passes through the three vertices of the triangle.
%
% The circumcircle contains the triangle, but it is not necessarily the
% smallest triangle to do so.
%
% If all angles of the triangle are no greater than 90 degrees, then
% the center of the circumscribed circle will lie inside the triangle.
% Otherwise, the center will lie outside the triangle.
%
% The circumcenter is the intersection of the perpendicular bisectors
% of the sides of the triangle.
%
% Surprisingly, the diameter of the circle can be found by solving
% a 2 by 2 linear system. If we label the vertices of the triangle
% P1, P2 and P3, then the vectors P2 - P1 and P3 - P1 are secants of
% the circle, and each forms a right triangle with the diameter.
% Hence, the dot product of P2 - P1 with the diameter vector is equal
% to the square of the length of P2 - P1, and similarly for P3 - P1.
% This determines the diameter vector originating at P1.
%
% In geometry, the circumcenter of a triangle is often symbolized by "O".
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 05 February 2005
%
% Author:
%
% John Burkardt
%
% Reference:
%
% Adrian Bowyer and John Woodwark,
% A Programmer's Geometry,
% Butterworths, 1983.
%
% Input:
%
% real T(2,3), the triangle vertices.
%
% Output:
%
% real CENTER(2,1), the circumcenter of the triangle.
%
dim_num = 2;
nrhs = 1;
%
% Set up the linear system.
%
a(1,1) = t(1,2) - t(1,1);
a(1,2) = t(2,2) - t(2,1);
a(1,3) = ( t(1,2) - t(1,1) ).^2 + ( t(2,2) - t(2,1) ).^2;
a(2,1) = t(1,3) - t(1,1);
a(2,2) = t(2,3) - t(2,1);
a(2,3) = ( t(1,3) - t(1,1) ).^2 + ( t(2,3) - t(2,1) ).^2;
%
% Solve the linear system.
%
[ a, info ] = r8mat_solve ( dim_num, nrhs, a );
%
% Compute the center
%
if ( info ~= 0 )
center(1:dim_num,1) = 0.0;
else
center(1:dim_num,1) = t(1:dim_num,1) + 0.5 * a(1:dim_num,dim_num+1);
end
return
end