perplexus.info :: Geometry : Misshapen Butter Pats

Misshapen Butter Pats (Posted on 2005-08-16)

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?

Submitted by McWorter

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Solution: (Hide)

A knife and a compass is enough. Let ABCD be a quadrilateral and suppose that AC is in the interior of ABCD (at least one diagonal of a quadrilateral must be interior). Draw a line through B parallel to AC and let B' be its intersection with AD. Then triangles ABC and AB'C have the same area. If that area is less or equal the area of triangle ACD, then the line CE that bisects B'D and E also lies between A and D. Hence CE bisects the area of quadrilateral ABCD. If the area of triangle ABC is greater than the area of triangle ACD, then repeat the construction using a line through D (instead of B) parallel to AC.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(4): Solution -- You're right; I'm wrong	McWorter	2005-08-18 01:36:10
re: Out Of The Box -> It Is A Box - A slip.. sideways	brianjn	2005-08-17 12:22:30
Out Of The Box -> It Is A Box	owl	2005-08-17 03:10:06
re(3): Solution -- You're right; I'm wrong	Bractals	2005-08-17 00:31:26
re(2): Solution -- You're right; I'm wrong	McWorter	2005-08-16 22:56:59
re: Solution	McWorter	2005-08-16 22:02:03
Solution	Bractals	2005-08-16 15:59:37
Manners	owl	2005-08-16 14:04:49
i hope so	Jud	2005-08-16 14:02:30

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