Trilinear coordinates

Trilinear coordinates (more precisely homogeneous trilinear coordinates) are in the triangular geometry, introduced by Julius Plücker tool to describe the position of a point with respect to a triangle.

Definition and notation

Given a triangle ABC. For any point P of the plane of the hot three real numbers, and (homogeneous ) trilinear coordinates of P, if there is a real number other than 0, so

Applies. This call, and the signed distances of the point P of the straight lines BC, CA and AB, respectively. The size receives a positive sign if P lies on the same side of BC as the corner A, and a negative sign if P and A are on different sides of BC. Accordingly, the other two signs are fixed.

The totality of the trilinear coordinates of a point is either written as an ordered triple or in the mold.

Trilinear coordinates are not uniquely defined: multiplication by an arbitrary real number not equal to 0 gives back trilinear coordinates of the given point.

Examples

  • The corners A, B and C of the given triangle have the trilinear coordinates, respectively.
  • The incenter of a triangle, the trilinear coordinates because it has the same distance from all three sides of the triangle.
  • For the centroid of a triangle, the trilinear coordinates loud or equivalent or. In this case, a, b, c of the side lengths and, for the sizes of the interior angle.

Connection with the barycentric coordinates

A simple relationship exists between the trilinear coordinates and barycentric coordinates are also frequently used in the triangle geometry: Are the trilinear coordinates given by, we obtain the barycentric coordinates, where a, b and c are the side lengths.

Formulas

Trilinear coordinates enable the use of algebraic methods in the triangle geometry in many cases. For example, three points, and the coordinates trilinear

Exactly collinear if the determinant

Is equal to 0. The dual to this set statement is also true: Three straight lines defined by the equations

Are given, then have exactly one point in common if and only if.

  • Triangle geometry
Inscribed circle
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