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Wikipedia / Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and .

In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinear coordinates to be non-positive.

Notation

for trilinear coordinates is often used in preference to the ordered triple represent the same point.

Examples

The trilinear coordinates of the incenter of a triangle are ; that is, the (directed) distances from the incenter to the sidelines are proportional to the actual distances denoted by , where is the inradius of . Given side lengths we have:Note that, in general, the incenter is not the same as the centroid; the centroid has barycentric coordinates (these being proportional to actual signed areas of the triangles , , , where = centroid.)

The midpoint of, for example, side for triangle area , which in arbitrarily specified relative distances simplifies to . The coordinates in actual sideline distances of the foot of the altitude from to which in purely relative distances simplifies to .

Formulas

Collinearities and concurrencies

are collinear if and only if the determinant equals zero. Thus if is a variable point, the equation of a line through the points and is . From this, every straight line has a linear equation homogeneous in in real coefficients is a real straight line of finite points unless is proportional to , the side lengths, in which case we have the locus of points at infinity.

concur in a point if and only if .

Also, if the actual directed distances are used when evaluating the determinant of , then the area of triangle is (and where is the area of triangle , as above) if triangle has the same orientation (clockwise or counterclockwise) as otherwise.

Parallel lines

are parallel if and only if where are the side lengths.

Angle between two lines

The tangents are given by

Thus they are perpendicular if

Altitude

The equation of the altitude from vertex to side is

Line in terms of distances from vertices

The equation of a line with variable distances from the vertices whose opposite sides are is

Actual-distance trilinear coordinates

The trilinears with the coordinate values being the actual perpendicular distances to the sides satisfy for triangle sides and area . This can be seen in the figure at the top of this article, with interior point partitioning triangle into three triangles , ,

Distance between two points

The distance between two points with actual-distance trilinears is given by

Distance from a point to a line

The distance from a point is

Quadratic curves

The equation of a conic section in the variable trilinear point is

It has no linear terms and no constant term.

The equation of a circle of radius having center at actual-distance coordinates is

Circumconics

The equation in trilinear coordinates of any circumconic of a triangle is

If the parameters respectively equal the side lengths (or the sines of the angles opposite them) then the equation gives the circumcircle.

Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center is

Inconics

Every conic section inscribed in a triangle has an equation in trilinear coordinates: with exactly one or three of the unspecified signs being negative.

The equation of the incircle can be simplified to while the equation for, for example, the excircle adjacent to the side segment opposite vertex can be written as

Cubic curves

Many cubic curves are easily represented using trilinear coordinates. For example, the pivotal self-isoconjugate cubic , as the locus of a point such that the -isoconjugate of is on the line

Among named cubics are the following:

Conversions

Between trilinear coordinates and distances from sidelines

For any choice of trilinear coordinates to locate a point, the actual distances of the point from the sidelines are given by , , where in which are the respective sidelengths , and is the area of .

Between barycentric and trilinear coordinates

A point with trilinear coordinates has barycentric coordinates where are the sidelengths of the triangle. Conversely, a point with barycentrics

Between Cartesian and trilinear coordinates

Given a reference triangle , express the position of the vertex in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector using vertex as the origin. Similarly define the position vector of vertex as Then any point associated with the reference triangle If this point has trilinear coordinates then the conversion formula from the coefficients and in the Cartesian representation to the trilinear coordinates is, for side lengths opposite vertices

More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors and if the point has trilinear coordinates , then the Cartesian coordinates of are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates as the weights. Hence the conversion formula from the trilinear coordinates to the vector of Cartesian coordinates

See also

References