You are asked to help design a Three Lock Box. Your job is to decide the locations of the three locks. The door is a unit square. To operate properly, each lock must be the same distance from the nearest door edge and also that same distance from each of the other two locks.

(In reply to re: solution by brianjn)

Maybe I can get this one right.

As each angle of an equilateral triangle is 60 degrees (pi/3 radians), we can determine the maximum the size of our triangle. Rotating it 15 degrees (pi/12 radians), that is, centering it so the bisector of the triangle is parallel to the diagonal of the square, we can calculate the distance the rotated triangle will be between the equidistant edges of the square. Let us assign d as the distance from the edge of the square and the length of the side of the triangle. Therefore, we have the two distances -- d and (1-2d) -- and the angles -- pi/2 and (pi/2 - pi/12 =) 5pi/12 -- needed to find the value of d.
Using the Law of Sines: 
     sin(5pi/12)/(1 - 2d) = sin(pi/2)/d
=> sin(5pi/12)/(1 - 2d) = 1/d
=> sin(5pi/12) = 1/d*(1 - 2d)
=> sin(5pi/12) = 1/d - 2
=> sin(5pi/12) + 2 = 1/d
=> d = 1/(sin(5pi/12) + 2)
=> d =~ 0.33716284848943383961467150932587

Edited on July 20, 2010, 2:13 pm

Posted by Dej Mar on 2010-07-20 10:56:24