Viviani's curve

Author:
RMF
Construction of Viviani's curve inspired by Villalpando's proportionatrix secunda (as Viviani claims) in a semicircle of diameter AC. Steps to find a point K of the curve from a point F of the semicircle:
  1. Semicircle of diameter AC (see diagram).
  2. A point F on the large semicircle is selected and FC drawn.
  3. FC is drawn.
  4. FL is perpendicular to AC.
  5. LK is perpendicular to CF.
  6. Unite AF.
The bottom bar of the canvas allows to follow these steps. Once the last step of the construction is arrived at, click on the 'Trace' button: the trajectory of the red point will trace the curve made by the point K when F moves on the semicircle. To stop the process, click on 'Stop Trace'. You can also draw the curve by selecting the point on the larger semicircle and moving it along the semicircle. To clean the canvas and get the initial configuration, click on the button with two circular arrows in the right upper corner of the canvas.
Once the curve is drawn, it is easy to prove that AC/CF = CF/CL = CL/CK because the figure ALCKF is precisely an Archytas' triangle. This statement is proved in Viviani's Diporto Geometrico, pp.279-280, http://www.e-rara.ch/zut/content/pageview/3800973. The two-dimensional curve described by Knorr is almost this same definition.