# Cevians $AD$, $BE$, $CF$ are concurrent, as are cevians $DP$, $EQ$, $FR$; show that $AP$, $BQ$, $CR$ are concurrent

The following is one version of the Cevian Nest Theorem:

In $$\triangle ABC$$, $$D$$, $$E$$, and $$F$$ are points on $$BC$$, $$CA$$, and $$AB$$, respectively, such that $$AD$$, $$BE$$, and $$CF$$ are concurrent lines. Points $$P$$, $$Q$$, and $$R$$ respectively on $$EF$$, $$FD$$, and $$DE$$ are such that $$DP$$, $$EQ$$, and $$FR$$ are concurrent. Prove that $$AP$$, $$BQ$$, and $$CR$$ are also concurrent.

I am really not getting how to proceed at all. I know I'm supposed to use Ceva's theorem but where do I apply it.

In the above picture, the three lines $$XI, YH, ZG$$ are concurrent if and only if $$\frac{YI}{IZ}\cdot \frac{ZH}{HX} \cdot\frac{XG}{GY} = 1.$$ Now, the ratios of the lengths are equal to the ratios of the areas of certain triangles, more specifically: $$\frac{YI}{IZ} = \frac{\triangle XYI}{\triangle XZI},\dots$$ here, we simply denote the triangle by its area. But $$\triangle XYI = \frac12 XY\cdot XI \cdot \sin (\angle YXI),$$ and similar for $$\triangle XZI$$. So $$\frac{YI}{IZ} = \frac{XY}{XZ}\cdot \frac{\sin\angle YXI}{\sin \angle ZXI}.$$ Taking cyclic product, we have the sine version of Ceva's theorem: That the three lines $$XI, YH, ZG$$ are concurrent if and only if the product $$\frac{\sin\angle YXI}{\sin \angle ZXI} \cdot \frac{\sin\angle ZYH}{\sin \angle XYH} \cdot \frac{\sin\angle XZG}{\sin \angle YZG}$$ of cyclic ratios of sines is equal to $$1$$.
So we want to show that $$\mathcal{F} = \frac{\sin\angle BAP}{\sin\angle PAC}\cdot \frac{\sin\angle ACR}{\sin\angle RCB} \cdot \frac{\sin\angle{CBQ}}{\sin\angle QBA} = 1.\tag{1}$$ Now $$\frac{\sin\angle BAP}{\sin\angle PAC} \cdot \frac{AF}{AE} =\frac{\triangle AFP}{\triangle AEP} = \frac{FP}{PE},$$ and so on. Taking the cyclic product of the above, we obtain $$\mathcal{F} \cdot \frac{AF}{AE}\cdot\frac{BD}{BF}\cdot \frac{CE}{CD} = \frac{FP}{PE}\cdot\frac{ER}{RD}\cdot\frac{DQ}{QF}.$$ And (1) follows from classic Ceva's theorem with the given concurrences.