GMT method

The GMT method (Geometric Mean Theorem method) is a problem-solving technique in geometry that uses the properties of similar triangles and the geometric mean to find unknown lengths, especially in problems involving right triangles and circles. It is often used to find loci or solve for points that satisfy certain distance conditions.

Important properties

• The geometric mean theorem states that in a right triangle, the altitude to the hypotenuse divides the hypotenuse into two segments, and the length of the altitude is the geometric mean of the lengths of these two segments.

• The GMT method often involves constructing auxiliary lines (such as altitudes or tangents) to reveal similar triangles.

• It is useful for finding loci of points that satisfy a product of distances condition, such as $PA \cdot PB = k^2$ for fixed points $A$ and $B$.