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 Divide in Half (Posted on 2014-03-07)

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

 Subject Author Date re(4): Response to Solution Bractals 2014-03-26 12:10:48 re(3): Response to Solution Harry 2014-03-25 21:32:12 re(2): Response to Solution Bractals 2014-03-19 12:22:21 re: Response to Solution Harry 2014-03-18 13:12:20 re: Response to Solution Harry 2014-03-18 12:14:54 Response to Solution Bractals 2014-03-17 19:39:38

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