<s>[INST] Area ratio in triangle? Given: $\triangle ABC$. In the side $AB$, we choose point $D$. From this point $D$, we draw a line $DF$ such that intersect side $AC$ and line $DE$ such that intersect side $BC$. If $DF\parallel BC$, $DE\parallel AC$, and the area of $\triangle BDE = p$ times area of $\triangle$ADF, what is the ratio of area $\triangle CEF$ and $\triangle ABC$ ?
 [/INST] The quadrilateral $CEDF$ is a parallelogram, and therefore $CE=DF=d$, $CF=DE=e$, and $\triangle CEF \equiv \triangle DEF$.

Suppose $h$ is the height of parallelogram $CEDF$ and $\ell$ is the height of $\triangle ADF$. Then, $h$ will also be the height of $\triangle BDE$ and $\triangle CEF$.
$$\therefore \ \text{Area of }\triangle CEF = \text{Area of }\triangle DEF = \frac12 \text{ Area of $CEDF$ parallelogram} =  \frac12 dh$$
As shown in the diagram (use your knowledge on parallelograms), you may find $\triangle BDE$ and $\triangle ADF$ are similar. Thus, their corresponding side lengths have the same ratio:
$$ \frac{BE}{DF} =  \frac{BD}{DA} =  \frac{DE}{AF} =  \frac{c}{d} =  \frac{b}{a} =  \frac{e}{f} \tag1$$
Suppose areas of $\triangle BDE$ and $\triangle ADF$ are $A_{\triangle BDE}$ and $A_{\triangle ADF}$, respectively. Given that:
$$\frac{A_{\triangle BDE}}{A_{\triangle ADF}} = p  \ \Rightarrow \ A_{\triangle ADF} = \frac{1}{p} A_{\triangle BDE} \tag2$$
$$\therefore \ \frac{\frac12ch}{\frac12d\ell} = \frac{ch}{d\ell} = p \ \Rightarrow \ \frac{c}{d} = \frac{\ell}{h}p \tag 3$$
From the equations $(1)$ and $(2)$, $\frac{b}{a} = \frac{\ell}{h}p$ and $\frac{e}{f} = \frac{\ell}{h}p$, as well.
You may also find $\triangle CGF$ and $\triangle AFH$ are similar as well. Thus, their corresponding side lengths have the same ratio:
$$ \frac{AF}{CF} =  \frac{AH}{FG} =  \frac{f}{e} =  \frac{\ell}{h}   \ \Rightarrow \ \frac{e}{f} = \frac{h}{\ell} $$
However, we previously found $\frac{e}{f} = \frac{\ell}{h}p$
$$\therefore \frac{\ell}{h}p = \frac{h}{\ell} \ \Rightarrow \ \frac{h^2}{\ell^2} = p \ \Rightarrow \ \frac{h}{\ell} = \sqrt{p} \tag4$$
$$\frac{A_{\triangle BDE}}{A_{\triangle CEF}} = \frac{\frac12ch}{\frac12dh} = \frac{ch}{dh} = \frac{c}{d} = \frac{\ell}{h}p = \frac{p}{\sqrt{p}} \ \Rightarrow \ \therefore \ A_{\triangle BDE} = \frac{p}{\sqrt{p}} A_{\triangle CEF} \tag5$$
From the equation $(2)$:
$$A_{\triangle ADF} = \frac{1}{p} A_{\triangle BDE} = \frac{1}{p} \frac{p}{\sqrt{p}} A_{\triangle CEF} = \frac{1}{\sqrt{p}} A_{\triangle CEF} \tag6$$
If the area of $\triangle ABC$ is $A_{\triangle ABC}$, then:
$$A_{\triangle ABC} = A_{\triangle ADF} + A_{\triangle BDE} + 2 \times A_{\triangle CEF} = \frac{1}{\sqrt{p}} A_{\triangle CEF}  + \frac{p}{\sqrt{p}} A_{\triangle CEF} + 2 \times A_{\triangle CEF} \\= \left(\frac{1}{\sqrt{p}} + \frac{p}{\sqrt{p}} + 2 \right)A_{\triangle CEF} $$
$$\therefore \ \frac{ A_{\triangle ABC}}{A_{\triangle CEF} } = \frac{\sqrt{p}}{p}(1+p) + 2$$
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