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Reference.org
Trilinear coordinates
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<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the trilinear coordinates <i>x</i> : <i>y</i> : <i>z</i> of a point relative to a given <a href="/facts/Triangle/cRXUwsMn">triangle</a> describe the relative <a href="/facts/Directed_distance/hbXlaEFk">directed distances</a> from the three <a href="/facts/Extended_side/wtqKRrBS">sidelines</a> of the triangle. Trilinear coordinates are an example of <a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinates</a>. The ratio <i>x</i> : <i>y</i> is the ratio of the <a href="/facts/Perpendicular_distance/P0aEswE4">perpendicular distances</a> from the point to the sides (extended if necessary) opposite <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a> A and B respectively; the ratio <i>y</i> : <i>z</i> is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for <i>z</i> : <i>x</i> and vertices C and A. </p><p>In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (a', b', c'), or equivalently in ratio form, <i>ka' </i>: <i>kb' </i>: <i>kc' </i> for any positive constant k. 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. </p>