The picture below on the left shows a right triangle JML with altitude KM. This altitude is defined as the geometric mean of JK and KL. Using similar triangles, you can prove that JK × KL = KM². Another way to think about
The picture below on the left shows a right triangle JML with altitude KM. This altitude is defined as the geometric mean of JK and KL. Using similar triangles, you can prove that JK × KL = KM². Another way to think about this construction is shown below on the right. Start with segments JK and KL placed end to end. Find the midpoint of JL and use it to construct a circle or semicircle with JL as its diameter. Construct a line through point K perpendicular to JL and mark where it intersects the semicircle at point M. Again, KM is the geometric mean of JK and KL since ∠JML (inscribed in a semicircle) is 90 degrees.
In Beautiful Geometry by Eli Maor and Eugen Jost, the authors use the geometric mean construction as a way to construct a square with area equal to a given rectangle. Using the semicircle construction of the geometric mean, they build a rectangle with side lengths JK and KL and a square with side length KM. Below is a picture from Beautiful Geometry showing the construction for several particular lengths.
When I saw the picture, I rushed to create a dynamic Web Sketchpad version of it. Below (and here on its own page), you’ll see my model. Press Animate to set the model in motion.
My interest in this problem dates back to 1990 when I was a graduate student in mathematics education at Cornell University taking a geometry course from David Henderson. David’s teaching philosophy was based on the Moore method, named after the topo