Determine the smallest positive real number
with the following property. Let
be a convex quadrilateral, and let points
,
,
, and
lie on sides
,
,
, and
, respectively. Consider the areas of triangles
,
,
and
; let
be the sum of the two smallest ones, and let
be the area of quadrilateral
. Then we always have
.
Author: unknown author, USA
%V0
Determine the smallest positive real number $k$ with the following property. Let $ABCD$ be a convex quadrilateral, and let points $A_1$, $B_1$, $C_1$, and $D_1$ lie on sides $AB$, $BC$, $CD$, and $DA$, respectively. Consider the areas of triangles $AA_1D_1$, $BB_1A_1$, $CC_1B_1$ and $DD_1C_1$; let $S$ be the sum of the two smallest ones, and let $S_1$ be the area of quadrilateral $A_1B_1C_1D_1$. Then we always have $kS_1\ge S$.
Author: unknown author, USA