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?

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