
Re: Classification of quadrilaterals
Posted:
Dec 21, 1994 5:06 PM


Yes, at least for convex quadrilaterals, it is true that if there is one bisector segment which bisects the quadrilateral, then the quadrilateral must be a trapezoid.
Sketch of proof:
Fix a line MM' (the besector segment), and draw AB with midpoint M.
Assume AB<= CD. Consider C'D' with midpoint M and length C'D'=length CD.
We ask when can area AC'M'M = area MM'D'B ?
Notice that area MM'A = area MM'B.
So equivalently, we ask when can area AM'C' = area BM'D' ?
In other words, when can the altitudes of the triangles AM'C' and BM'D' as measured from the base on C'D' be equal? This occurs when AB  C'D'.
I bet this can be modified to accomodate nonconvex quadrilaterals too... but alas! I've a plane to catch...darn! just when things are getting interesting...
Happy Holidays everyone, Ken
A C'
M M'
B D'

