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On Target (Posted on 2006-10-09) Difficulty: 4 of 5
At the beginning of the exercise, three soldiers, named Ike, Jay and Kay, were at three different points equidistant from a target. Ike was 4 kilometers from Jay, and also 4 kilometers from Kay.

Then Jay started moving inward, directly toward the target. He stopped short of the target, at a point different from his original location, but again 4 kilometers from Ike.

At this point the distances between any two of Ike, Jay, Kay and the target were all whole numbers of kilometers.

In his new position, how far is Jay from Kay?

See The Solution Submitted by Charlie    
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Solution Solution | Comment 4 of 6 |

I assumed that this is a two dimensional problem.
Let the soldier's initial positions be represented
by the first letter of their names. The points I,
J, and K lie on a circle with center the target
(point T) and radius R km. For a solution R > 4.
Let J' be the position of Jay after moving inward.
The points J, K, and J' lie on a circle with center
I and radius 4 km. Triangles TIJ and IJJ' are
similar. Thus,
  TI     IJ         IJ
 ---- = ----- = ----------
  IJ     IJ'     JT - J'T
         R^2 - 16
  J'T = ----------                        (1)
Quadrilateral TKIJ' is cyclic. Thus,
  (J'K)(TI) = (J'T)(KI) + (TK)(IJ')
         R(J'K - 4)
  J'T = ------------                      (2)
Combining (1) and (2) gives
  J'K = 8 - (8/R)^2
Therefore, J'K ranges from 4 km to 8 km
depending on the value of R.

  Posted by Bractals on 2006-10-09 20:59:54
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