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

[list=1][*]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.[/*][/list]