Pech-Blažek theorem

A recent study by Prof. Pavel Pech and his student Jiří Blažek obtained a nice result on a necessary condition for the point P to be placed if its projections on the sides of a quadrilateral define two parallel lines p and q. They found that GeoGebra can investigate the situation by visualizing a subset of the possible points (see the magenta curve in the figure).


  1. Investigate the algebraic equation of the magenta curves while you change the vertices of the quadrilateral. Which order do the equations usually have?
  2. Classify the obtained algebraic equations in the order you have generally.
  3. You may try to obtain further information about the obtained curves by explicitly finding their parameters, but this is a difficult task. Also, when finding the exact curve, your computer may be too slow to explicitly prove this theorem by using GeoGebra.