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Topic: Classification of quadrilaterals
Replies: 17   Last Post: Jan 11, 1995 6:43 AM

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Chenteh Kenneth Fan

Posts: 1
Registered: 12/10/04
Re: Classification of quadrilaterals
Posted: Dec 21, 1994 5:06 PM
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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 non-convex 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'








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