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).
Challenges
- Investigate the algebraic equation of the magenta curves while you change the vertices of the quadrilateral. Which order do the equations usually have?
- Classify the obtained algebraic equations in the order you have generally.
- 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.