Inequality on a Quadrilateral
Source: 1989 IrMO Paper 1 Problem 1
September 29, 2017
geometryinequalities
Problem Statement
A quadrilateral is inscribed, as shown, in a square of area one unit. Prove that
[asy]
size(6cm);
draw((0,0)--(10,0));
draw((10,0)--(10,10));
draw((0,10)--(10,10));
draw((0,0)--(0,10));dot((0,8.5)); dot((3.5,10)); dot((10,3.5)); dot((3.5,0));
label("",(0,8.5),W);
label("",(3.5,10),NE);
label("",(10,3.5),E);
label("",(3.5,0),S);
draw((0,8.5)--(3.5,10));
draw((3.5,10)--(10,3.5));
draw((10,3.5)--(3.5,0));
draw((3.5,0)--(0,8.5));
[/asy]