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
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