We'll return back to Sampy's question later. But, for a change let's see a simple one - or at least it looks simple. We're looking for the shortest proof of the claim below. May be purely-geometric one? Or is the claim wrong? Let us know...

Take trapezoid ABCD with BC // AD and |BC|<|AD|. Let E on AB be the point such that area(ADE)=area(EBC). Parallels to BC in E and to AB in C cross in F. By construction, F is inside triangle CDE and area(ECF)=area(EBC). Therefore area(CDE)>area(EBC).

Precision: problem claim is true provided that the three triangular pieces share a vertex on one of the trapezoid's leg as hinted in the problem picture. If the common vertex is on the longest base we can find obvious counterexamples.

Thank you for this simple -and possibly the simplest- solution.

Admittedly, we don't always pay deserving attention to problem statements here. In those cases, pictures should complement text. Otherwise, please do raise your voice. Sorry if that led to wrong conclusions.

Also, British English definition of Trapezoid is news to me. Isn't learning a beautiful thing. It should never end, no, no, should never ever end.

http://bleaug.free.fr/8foxes/8foxes348.png

ReplyDeleteTake trapezoid ABCD with BC // AD and |BC|<|AD|. Let E on AB be the point such that area(ADE)=area(EBC). Parallels to BC in E and to AB in C cross in F. By construction, F is inside triangle CDE and area(ECF)=area(EBC). Therefore area(CDE)>area(EBC).

bleaug

A trapezoid has no parallel sides! jyu

ReplyDeleteThe claim is wrong! jyu

ReplyDeletehttp://en.wikipedia.org/wiki/Trapezoid

ReplyDeletebleaug

Precision: problem claim is true provided that the three triangular pieces share a vertex on one of the trapezoid's leg as hinted in the problem picture. If the common vertex is on the longest base we can find obvious counterexamples.

ReplyDeletebleaug

British English: A trapezoid has no parallel sides.

ReplyDeleteSo the claim is wrong if you adopt the British English version.jyu

ReplyDeleteThank you for this simple -and possibly the simplest- solution.

ReplyDeleteAdmittedly, we don't always pay deserving attention to problem statements here. In those cases, pictures should complement text. Otherwise, please do raise your voice. Sorry if that led to wrong conclusions.

Also, British English definition of Trapezoid is news to me. Isn't learning a beautiful thing. It should never end, no, no, should never ever end.

More to come later...

Thanks all very much.

-8foxes