Olympiade Math – Géométrie 01 – Ex 28

* Traingle DFC:
S=DF.DC /2
S=y.DC/2
➝ DC=2S/y

* Traingle BCE:
S=BC.BE /2 & BC=x+y
➝ BE=2S/(x+y)

* Traingle AFE:
AB=DC=2S/y
AE=AB-BE
AE=2S/y-2S/(x+y)
AE=2Sx/y(x+y)
d’où:
S’=AExAF/2
S’=2Sx²/2y(x+y)
➝ S’=Sx²/y(x+y)

* Rectangle ABCD
SABCD=3S+S’=ADxDC
3S + Sx²/y(x+y) = (x+y)2S/y
3 + x²/y(x+y) = 2(x+y)/y 
3y(x+y)+x²=2(x+y)²
3xy+3y²+x²=2x²+4xy+2y²
y²-xy-x²=0 (1/x²)
(y/x)²-(y/x)-1=0
on pose: t=y/x
➝ t²-t-1=0