A linear combination of points or 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 points or vectors.
Definition: The sum
SUM_{j}[ c_{j}] |
of the coefficients in a linear combination is called the weight of the linear combination.
A barycentric combination of points or vectors v_{1}, ..., v_{N} is any weight 1 linear combination of them in which each coefficient is non-negative.
If A and B are two different points of the plane (or of space or of n-dimensional space), then the line determined by A and B is the set of all weight 1 linear combinations of A and B, and the line segment between A and B is the set of all barycentric combinations of A and B. Note that if V = B - A is the vector from A to B, then the line determined by A and B is the set of all points A + tV, and the line segment AB is the subset of these points with 0 <= t <= 1.
Theorem. If A, B, and C are any non-collinear points in the Cartesian plane, then every point X of the plane is a unique weight 1 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 in the triangle determined by the three points if and only if it is a barycentric combination of A, B, and C.
Let A, B, C, and D be the points in the Cartesian plane that are given by
A = (0, -1) , B = (3, 4) , C = (-1, 1) , and D = (1, 2) , |
and let T be the triangle with vertices A, B, and C.
Find the midpoint of the line segment AB.
For which values of t does the point (1-t)A + tB lie on the line segment AB ?
Find the point where the line AC meets the line BD. Does this intersection point lie on both of the line segments AC and BD ?
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 barycentric coordinates of the point (2, 2) with respect to the vertices of the triangle T.
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