In the figure, the yellow octagon
B_{1}B_{2}...B_{8} is the midpoint octagon
of
an octagon A_{1}A_{2}...A_{8}. The red octagon
C_{1}C_{2}...C_{8} is the midpoint octagon
of
the yellow octagon B_{1}B_{2}...B_{8.} O_{1}, O_{2},
..., O_{8} are the centers of yellow squares on side B_{1}B_{2}, ... B_{8}B_{1}. Q_{1}, Q_{2},
..., Q_{8} are the centers of red squares on sides C_{1}C_{2}, ..., C_{8}C_{1}.
O_{15}, O _{26}, O _{37}, O _{48}, Q_{15}, Q_{26}, Q_{37},
and Q_{48} are midpoints of O_{1}O_{5}, O_{2}O_{6}, O_{3}O_{7}, O _{4}O_{8}, Q_{1}Q_{5}, Q_{2}Q_{6}, Q_{3}Q_{7},
and Q_{4}Q_{8}, respectively. Prove that (1)
Quadrilateral O_{15}O_{26}O_{37}O_{48} and Q_{15}Q_{26}Q_{37}Q_{48} are squares (2) Q_{15}, Q_{26}, Q_{37},
and Q_{48} are the midpoints of
O_{15}O_{26}, O_{26}O_{37}, O_{37}O_{48}, and O_{48}O_{15}, respectively.

**The midpoint polygon** of a polygon P is the polygon whose vertices are the midpoints of the
sides of P.

Geometric art is a form of art based on the use and application of geometric figures. A geometric figure is any set or combination of points, lines, surfaces and solids. A mobile app or mobile application software is a computer program designed to run on smartphones and tablet computers.