While buttering my bread at lunch one day I mused that one can not only cut a square pat of butter in half with only a knife, but also if the pat is in the shape of a regular ngon, n>3. But what if the pat is shaped like an irregular quadrilateral?
Can you bisect the area of an arbitrary quadrilateral with one straight line using only a straightedge and a compass?
Label the quadrilateral ABCD and construct line AC.
Construct a line through B perpendicular to AC and
intersecting AC at B'. Construct a line through D
perpendicular to AC and intersecting AC at D'.
WOLOG let BB' >= DD'. Construct point G on line
segment BB' such that
BG^2 = (BB' + DD')(BB'/2).
The line through G and parallel to AC is the
desired cut.
Note: If the quadrilateral is not convex,
label so that C is the vertex with interior
angle greater than 180 degrees.
Edited on August 16, 2005, 4:07 pm

Posted by Bractals
on 20050816 15:59:37 