KöMaL 2021 A799

For a given quadrilateral $A_1A_2B_1B_2$ point $P$ is called phenomenal, if line segments $A_1A_2$ and $B_1B_2$ subtend the same angle at point $P$ (i.e. triangles $PA_1A_2$ and $PB_1B_2$ which can be also also degenerate have equal inner angles at point $P$ disregarding orientation). Three non-collinear points, $A_1$, $A_2$ and $B_1$ are given on the plane. Prove that it is possible to find a disc on the plane such that for every point $B_2$ on the disc quadrilateral $A_1A_2B_1B_2$ is convex for which it is possible to construct seven distinct phenomenal points only using a right ruler. With a right ruler the following two steps are allowed: $i)$ given two points it is possible to draw the straight line connecting them; $ii)$ given a point and a straight line, it is possible to draw the straight line passing through the given point which is perpendicular to the given line.