Geometric mean theorem

In Euclidean geometry, the geometric mean theorem or right triangle altitude theorem is a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude.

History[edit]

The theorem is usually attributed to Euclid (ca. 360–280 BC), who stated it as a corollary to proposition 8 in book VI of his Elements. In proposition 14 of book II Euclid gives a method for squaring a rectangle, which essentially matches the method given here. Euclid however provides a different slightly more complicated proof for the correctness of the construction rather than relying on the geometric mean theorem.^[1]^[3]

consider triangles $△ ABC, △ ACD$ ; here we have
$\angle ACB=\angle ADC=90^{\circ },\quad \angle BAC=\angle CAD;$ therefore by the $\triangle ABC\sim \triangle ACD.$

AA postulate

further, consider triangles $△ ABC, △ BCD$ ; here we have
$\angle ACB=\angle BDC=90^{\circ },\quad \angle ABC=\angle CBD;$ therefore by the AA postulate $\triangle ABC\sim \triangle BCD.$

at Cut-the-Knot

Geometric mean theorem

History[edit]

AA postulate

Geometric Mean