GMT with non-zero area

In geometry, the locus of points that form triangles (or other polygons) with a given non-zero area with respect to fixed points is an important concept. For example, given two fixed points $A$ and $B$, the locus of points $P$ such that the area of triangle $ABP$ is a fixed non-zero value $S$ forms two lines parallel to $AB$, one on each side of $AB$. This is because the area of a triangle can be expressed as $S = \frac{1}{2} \cdot |AB| \cdot h$, where $h$ is the distance from $P$ to line $AB$. Thus, all points $P$ at a fixed perpendicular distance from $AB$ (but not on $AB$) will satisfy the area condition.

Important properties

• The locus is usually a curve or set of curves (often lines or circles) depending on the area condition and the fixed points.

• For triangle area with two fixed points, the locus is two lines parallel to the segment joining the fixed points.

• The locus excludes degenerate cases (e.g., area zero, which would place $P$ on the line through $A$ and $B$).