The figure shows a hexagon
A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}
with equilateral triangles
A_{1}A_{2}B_{1},... A_{6}A_{1}B_{6}.of
centers O_{1}, O_{2}, ..., O_{6}. Q_{1}, Q_{2}, ...,Q_{6} are the centers of equilateral
triangles B_{1}B_{2}C_{1}, ..., B_{6}B_{1}C_{6}. O_{14}, O_{25}, O_{36}, Q_{14}, Q_{25}, and Q_{36}
are the midpoints of O_{1}O_{4}, O_{2}O_{5}, O_{3}O_{6}, Q_{1}Q_{4}, Q_{2}Q_{5}, and Q_{3}Q_{6},
respectively. Prove that (1) Triangles O_{14}O_{25}O_{36} and Q_{14}Q_{25}Q_{36}
are equilateral; (2) O_{14}, O_{25}, and O_{36} are the midpoints of
Q_{14}Q_{36}, Q_{14}Q_{25}, and Q_{25}Q_{36}, respectively.
See also:
Sketch of problem 1319 using mobile apps
