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\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}, areas inside quadrilateral

Source: Norwegian Mathematical Olympiad 2001 - Abel Competition p3b

February 22, 2020

Geometric Inequalitiesinequalitiesareatriangle areageometry

Problem Statement

The diagonals

AC

and

BD

in the convex quadrilateral

ABCD

intersect in

S

. Let

F_1

and

F_2

be the areas of

\vartriangle ABS

and

\vartriangle CSD

. and let

F

be the area of the quadrilateral

ABCD

. Show that

\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}