Given an arbitrary convex quadrilateral, construct a triangle of equal area using only straight edge and...

geometry/construction/quadrilateral.to.triangle New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Given an arbitrary convex quadrilateral, construct a triangle of equal
area using only straight edge and compass.

-- Thomas Gray <ai380@FreeNet.Carleton.CA>

Show solution

Label the vertices consecutively A, B, C, and D.
AC is a diagonal of ABCD.
Construct EF parallel to AC passing through B.
Extend CD until it intersects EF, label the intersection B'.
Then AB'D is a triangle with the same area as ABCD.

Proof:

ABC and AB'C have the same base, AC.
ABC and AB'C have the same height, the distance between AC and EF.
Therefore, area(ABC) = area(AB'C).
area(ABCD) = area(ABC) + area(ACD).
area(AB'D) = area(AB'C) + area(ACD).
Therefore, area(ABCD) = area(AB'D) [Editor: printed as area(AB'C)].
QED

-- Dave Dodson <dodson@convex.COM>