Perpendicular bisector and GMT

The perpendicular bisector of a segment $AB$ is the locus of all points equidistant from $A$ and $B$. The Geometric Mean Theorem (GMT), also called the Right Triangle Altitude Theorem, states that in a right triangle, the altitude to the hypotenuse divides the triangle into two smaller triangles that are similar to each other and to the original triangle. The length of the altitude is the geometric mean of the segments it divides the hypotenuse into.

Important properties

• Any point on the perpendicular bisector of $AB$ is equidistant from $A$ and $B$.

• The perpendicular bisector can be constructed using a compass and straightedge.

• The GMT relates the altitude to the hypotenuse in a right triangle: if the hypotenuse is divided into segments of lengths $p$ and $q$ by the altitude, then the altitude $h$ satisfies $h^2 = pq$.

• The intersection point of the perpendicular bisectors of a triangle is the circumcenter, which is equidistant from all three vertices.