Holfeld's 35th problem as an implicit locus
Ioannis Holfeld's Exercitationes Geometricae (1773, Prague) introduced the following problem:
Given a circle with center A and circumpoint M, radius r. Let B another circumpoint and C a point on line AB. What is the locus of points C such that MO/AO=r/BC?
Holfeld's solution was a parabola, shown below, but by using modern algebraic geometry methods and automatic computations another solution will also be delivered. Notably, a pretzel curve.
(For the details see R. Hašek, J. Zahradník: Contemporary interpretation of selected historical problems on loci. Proceeding of the 33rd Conference on Geometry and Graphics. Horní Lomná, September 9–12, 2013. See also R. Hašek, Z. Kovács, J. Zahradník: Loci problems in Age of Reason and their effect on GeoGebra: GeoGebra's contribution to solving loci problems, Locus equations and their factorization in GeoGebra, presentations at the International GeoGebra Conference, Budapest, January 24, 2014.)
The pretzel curve is introduced because by using computational algebraic geometry we cannot prescribe the exact position of point C: it can be on different rays on the line AB.
In this applet we used GeoGebra's LocusEquation command. By right-clicking on the red curve, the exact syntax of the command can be read off.