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perimeter inequality in convex hegaxon , 1/2 \le s/(u+v) \le 1

Source: Austrian Regional Competition For Advanced Students 2002 p3

January 11, 2020

geometryperimeterinequalitiesconvexhexagongeometric inequality

Problem Statement

In the convex

ABCDEF

(has all interior angles less than

180^o

) with the perimeter

s

the triangles

ACE

and

BDF

have perimeters

u

and

v

respectively. a) Show the inequalities

\frac{1}{2} \le \frac{s}{u+v}\le 1

b) Check whether

1

is replaced by a smaller number or

1/2

by a larger number can the inequality remains valid for all convex hexagons.