A linear combination of vectors v_{1}, ..., v_{N} is any point of the form
SUM[_{j} c_{j} v_{j} ]
where the c_{j} are numbers. The numbers c_{j} are called the coefficients of the linear combination of the given vectors.A barycentric combination of vectors v_{1}, ..., v_{N} is any linear combination in which the sum of the coefficients is 1.
If A and B are two different points of the plane (or of space or of n-dimensional space), then the line segment determined by A and B is the set of all barycentric combinations of A and B.
Theorem. If A, B, and C are any non-collinear points in the Cartesian plane, then every point X of the plane is a barycentric combination of A, B, and C. .
If A, B, and C are any non-collinear points in the Cartesian plane, then a point X in the plane lies inside the triangle determined by the three points if and only if its barycentric coordinates with respect to A, B, C all lie in the unit interval 0 <= t <= 1.
Let A, B, and C be the points in the Cartesian plane that are given by
A = (0, -1) B = (3, 4) and C = (-1, 1) ,
and let T be the triangle with vertices A, B, and C.
Find the point where the three medians of T meet.
Find the point where the three perpendicular bissectors of the sides of T meet.
Find the point where the three angle bissectors of T meet.
Which of the above points are inside the triangle?
How does one decide by analytic methods if a point lies inside a triangle?
Find the barycentric coordinates of the point (5, 0) with respect to the vertices of the triangle T.
Find the barycentric coordinates of the point (2, 2) with respect to the vertices of the triangle T.
What properties of a triangle determine when any of the three canonical intersection points of the triangle, as above, does not lie inside the triangle?