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Divide in Half (Posted on 2014-03-07) Difficulty: 3 of 5

  
Given a convex quadrilateral. Construct, with straight-edge and
compass, a line parallel to one of the sides of the quadrilateral
that divides the area of the quadrilateral in half.
  

  Submitted by Bractals    
Rating: 5.0000 (1 votes)
Solution: (Hide)

  
There are three cases.

Case I: The quadrilateral is a parallelogram. Pick any side and
construct the desired line parallel to it and through the intersection
of the diagonals.

Case II: The quadrilateral is a trapezoid. Label the quadrilateral
ABCD such that AB || CD and point Y is the intersection of rays
AD and BC. Let PQ be the desired line with points P and Q on rays
YA and YB respectively.

     Area(ABQP) = Area(PQCD)  ⇔

         2|YP||YQ| = |YA||YB| + |YC||YD|       (1)

     PQ || AB  ⇔  |YA||YQ| = |YB||YP|         (2)
Eliminating |YP| from (1) and (2) gives

     |YQ| = sqrt( |YB|*(|YB| + |YC||YD|/|YA|)/2 )  

     |EF| = |YC||YD|/|YA|        is constructible
     |GH| = |YB| + |EF|          "  "
     |IJ| = |GH|/2               "  "
     |YQ| = sqrt(|YB||IJ|)       "  "
     PQ || AB                    "  "
Case III: No sides of the quadrilateral are parallel. Label the
quadrilateral ABCD such that point Y is the intersection of rays AD
and BC and point X is the intersection of rays AB and DC. Let PQ be
the desired line with points P and Q on rays YA and YB respectively.
The construction is identical to Case II.

The construction fails if |YQ| < |YC|. In that case let PQ || AD
be the desired line with points P and Q on rays XA and XD respectively.

     Area(ADQP) = Area(PQCB)  ⇔

         2|XP||XQ| = |XA||XD| + |XB||XC|       (3)

     PQ || AD  ⇔  |XA||XQ| = |XD||XP|         (4)
Eliminating |XP| from (3) and (4) gives

     |XQ| = sqrt( |XD|*(|XD| + |XB||XC|/|XA|)/2 )  

     |EF| = |XB||XC|/|XA|        is constructible
     |GH| = |XD| + |EF|          "  "
     |IJ| = |GH|/2               "  "
     |XQ| = sqrt(|XD||IJ|)       "  "
     PQ || AD                    "  "
The construction fails if |XQ| < |XC|.

See my "Response to Solution".

QED
  

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(4): Response to SolutionBractals2014-03-26 12:10:48
Some Thoughtsre(3): Response to SolutionHarry2014-03-25 21:32:12
re(2): Response to SolutionBractals2014-03-19 12:22:21
re: Response to SolutionHarry2014-03-18 13:12:20
re: Response to SolutionHarry2014-03-18 12:14:54
Response to SolutionBractals2014-03-17 19:39:38
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