Note that this open investigation can be taken to many levels of complexity

Draw a quadrilateral and label its vertices $A, B, C$ and $D$. Pick any point $P_0$ outside the quadrilateral. Apply the following operations to generate eight new points:

• Let $P_1$ be the point half way between $P_0$ and $A$
• Let $P_2$ be the point half way between $P_1$ and $B$
• Let $P_3$ be the point half way between $P_2$ and $C$
• Let $P_4$ be the point half way between $P_3$ and $D$
• Let $P_5$ be the point half way between $P_4$ and $A$
• Let $P_6$ be the point half way between $P_5$ and $B$
• Let $P_7$ be the point half way between $P_6$ and $C$
• Let $P_8$ be the point half way between $P_7$ and $D$

Explore the properties of this sequence and its continuation. You might like to use geogebra or other tools alongside a pen and paper analysis.

For more investigations see our Stage 5 pages.