The vertex centroid of a Van Aubel result involving similar quadrilaterals and its further generalisation

Figures & data

Figure 1. Quadrilateral with directly similar quadrilaterals on the sides.

Figure 2. The polygons $P_1{P}_2\dots P_n$ and $Q_1{Q}_2\dots Q_n$ share the same vertex centroid ($V)$

Figure 3. Angle ${\alpha }_2$ between $P_1{Q}_1$ and $P_2{Q}_2$ (extended).

Figure 4. Two directly similar 7-gons erected on consecutive sides of the initial polygon.