Becker's curve

Author:
RMF
Construction of Becker's curve in a semicircle of diameter AG. Steps to find a point K of the curve from a point F of the semicircle:
  1. Semicircle of diameter AG (see diagram).
  2. A point F on the large semicircle is selected and FG.
  3. FL is perpendicular to AG.
  4. Semicircle of diameter AL.
  5. K is the intersection of AF with the semicircle AL.
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 AG/AF = AF/AL = AL/AK: it is enough to show that triangles AGF and ALK and semicircles form an Archytas' triangle. Therefore, AKFGL is an Archytas' solution. Becker's curve is the same curve as Viviani (and Knorr), but its definition is different. That definition is the most fitted to the Archytas' construction, and Becker was aware of the fact that it "directly ties in with the planimetrical core piece of solution of Archytas".