<s>[INST] Euclidean geometry : find out the area. 
${\angle}ABC=45°$, $AD{\bot}AE$, $AD=AE$, $AC{\bot}BE$, $F$ is the intersection of $AC$ and $BE$, $BF=9$, so what is the area of ${\square}ABCE$?
 [/INST] Let $F(0,0), A(0,a), B(-b,0), C(0,-c), E(e,0)$, then $D=tB+(1-t)C=
(-tb,-(1-t)c)$ for some $t$ ($t\in(0;1)$ if we believe the drawing, but I won't use it).
The desired area is then $\frac12(ae+ec+bc+ba)$.
We have:
$AD\perp AE\Leftrightarrow (D-A).(E-A)=0 \Leftrightarrow (-tb,-(1-t)c-a).(e,-a)=0 \Leftrightarrow -tbe+(1-t)ca+a^2=0$
$|AD|=|AE|\Leftrightarrow (D-A)^2=(E-A)^2 \Leftrightarrow t^2b^2+(1-t)^2c^2+a^2+2(1-t)ca=a^2+e^2$
$AC\perp BE$ ve have already used when introducing the coordinate system
$BF=9\Leftrightarrow b=9$
and $\angle ABC=45^\circ \Leftrightarrow (A-B).(C-B)=\frac{\sqrt{2}}{2}|A-B|\cdot|C-B|\Rightarrow ((a,b).(b,-c))^2=\frac{1}{2}(a^2+b^2)(b^2+c^2)
\Leftrightarrow 2(ab-bc)^2=(a^2+b^2)(b^2+c^2)$
Then we wolframalpha this thing $\begin{cases}
-tbe+(1-t)ca+a^2=0\\
t^2b^2+(1-t)^2c^2+a^2+2(1-t)ca=a^2+e^2\\
b=9\\
2(ab-bc)^2=(a^2+b^2)(b^2+c^2)
\end{cases}$
and see there's one condition missing as the desired area $S=a^2 - \frac{2916}{a - 9} - 243$ can be made arbitrary large if we take $a$ close enough to $9$.
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