Bisection plane and GMT

A bisection plane is a plane that divides a segment or the space between two points into two equal parts. In 3D geometry, the set of all points equidistant from two fixed points $A$ and $B$ forms a plane called the perpendicular bisector plane of $AB$. GMT (Geometric Mean Theorem) in 3D often refers to loci involving distances, such as the locus of points with a constant ratio of distances to two fixed points.

Important properties

• The bisection plane of segment $AB$ is perpendicular to $AB$ and passes through its midpoint.

• Any point on the bisection plane is equidistant from $A$ and $B$.

• The locus of points $P$ such that $\frac{PA}{PB} = k$ (for constant $k > 0$) is a sphere (for $k \neq 1$) or the bisection plane (for $k = 1$).