In the picture all of the red segments are of equal length and all of their endpoints lie on one of the two blue lines. Determine all possible values for the smaller angle between the blue lines.

(This problem is a special case of a problem discovered by Daniel Shapiro, professor of mathematics at Ohio State University. His problem generalizes a problem he saw years ago.)

(In reply to re: Label the figure by McWorter)

Let x=COB = FOE
Let y=ACB = DFE = CAB = FDE
Let each red segment equal 1 = AB,BC,CD,DE,EF,FA
then CBO = y-x and BD=2cos(y-x)
also ABO = 180-x-y and BF = 2cos(180-x-y) = -2cos(x+y)
FD = 2sin(90-y) = 2cosy
and since BF+FD=BD
-2cos(x+y) + 2cosy = 2cos(y-x)
-cos(x+y) + cosy = cos(y-x)
-cosxcosy + sinxsiny + cosy = cosycosx + sinysinx
-cosxcosy + cosy = cosycosx
cosy = 2cosxcosy
and as long as cosy is not 0 (y isn't 90)
1 = 2cosx
x = Arccos 1/2 = 60

Edited on June 9, 2005, 4:00 pm

Posted by Eric on 2005-06-09 15:58:17