In a blue square A_{1}A_{2}A_{3}A_{4}
of side a, the trisectors of angles A_{1}, A_{2}, A_{3}, A_{4} and
diagonals intersect at the points B_{1}, B_{2}, B_{3}, B_{4}, C_{1}, C_{2}, C_{3}, C_{4}, D_{1}, D_{2}, D_{3}, ..., D_{12}
as shown in the figure.
Prove that (1) B_{1}B_{2}B_{3}B_{4} is a
square (yellow);
(2) D_{1}D_{4}D_{7}D_{10} is a square
(red); (3) C_{1}C_{2}C_{3}C_{4} is a square
(orange) ; (4) D_{1}D_{2}...D_{12} is a regular dodecagon
(green); (5) Area
B_{1}B_{2}B_{3}B_{4} =
; (6)
Area B_{1}B_{2}B_{3}B_{4} = 2 Area D_{1}D_{4}D_{7}D_{10};
(7) Area B_{1}B_{2}B_{3}B_{4} = 3 Area C_{1}C_{2}C_{3}C_{4};
(8) Area
D_{1}D_{2}...D_{12} = 3/4 Area B_{1}B_{2}B_{3}B_{4}.

See also:

Geometry Problems

Ten problems: 1321-1330

Visual Index

Open Problems

All Problems

Triangle

Equilateral Triangle

Square

Areas

30, 60 Angle

Angles

Problem 1357: Regular Dodecagon, Concurrency, Collinearity

Problem 1358: Square, Regular Dodecagon

Kurschak's Dodecagon