<s>[INST] Find the area of a quadrilateral in a parallelogram. 
Assume quadrilateral $ABCD$ is a parallelogram and its area is $S$. And satisfy the following conditions: $AE=BE$, $BF=FC$, $AQ // PC$. Find the area of quadrilateral $APCQ$
 [/INST] Assume the origin of the coordinate system is $B$. so coordinate of point $B$ is $(0,0)$. And the $x$-axes is $BC$.  Assume the coordinate of point $A$ is $(2a,2b)$. The coordinate of point $C$ is $(2c,0)$. Then we can calculate other points' coordinates. $E$ is $(a,b)$, $F$ is $(c,0)$ $D$ is $(2c+2a, 2b)$. 
Then the equation of straight line $AF$ is $$y=\frac{-2b}{c-2a}(x-c)$$
the equation of straight line $ED$ is $$y-b=\frac{b}{2c+a}(x-a)$$
so the coordinate of point $P$ is $(\frac{2}{5}(3a+c),\frac{6}{5}b)$
then the equation of straight line $AQ$ is $$y-2b=\frac{3b}{3a-4c}(x-2a)$$
then the coordinate of point $Q$ is $(\frac{4c+7a}{5},\frac{7}{5}b)$
then $$|PQ|=\sqrt{\left(\frac{b}{5}\right)^2+\left(\frac{a}{5}+\frac{2c}{5}\right)^2}$$
and $$|ED|=\sqrt{b^2+(2c+a)^2}$$
then we know $|ED| = 5\times |PQ|$ 
then $S_{APCQ} = \frac{1}{5}S_{AECD}$ Assume the area of $ABCD$ is $S$. the area of $AECD$ is $\frac{3}{4}S$. So $S_{APCQ}= \frac{3}{4}S \times \frac{1}{5} = \frac{3}{20}S$
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