In a convex quadrilateral all sidelengths and all angles are pairwise different. a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side? b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side?
Problem
Source: Sharygin Geometry Olympiad 2012 - Problem 10
Tags: geometry unsolved, geometry
yunxiu
02.05.2012 04:36
a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side? YES, there exit such convex quadrilateral.
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yunxiu
02.05.2012 05:45
b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side? Without loss of generality we suppose $\angle D$ is greatest angle，$AB$ is smallest side。 Compare $\Delta ADC,\Delta ABC$, because $AD > AB$, $\angle ADC > \angle ABC$, and $AC = AC$, we have $BC > DC$, so the greatest side is $AD$ or $BC$. If $AD$ is the greatest side. Compare $\Delta ADC,\Delta CBA$, because $AD > CB$, $\angle ADC > \angle CBA$, $DC > BA$, we have $AC > CA$, contradiction. If $BC$ is the greatest side, then $\angle BAD$ is the smallest angle. Compare $\Delta BCD,\Delta DAB$, $BC > DA$, $CD > AB$, $\angle DCB > \angle BAD$, so we have $DB > BD$, contradiction. Above all, there not exit such convex quadrilateral.
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