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Misshapen Butter Pats (Posted on 2005-08-16) Difficulty: 3 of 5
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 n-gon, 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?

See The Solution Submitted by McWorter    
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Solution Solution | Comment 3 of 9 |
 
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 2005-08-16 15:59:37

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