In $\triangle ABC$, consider points $D$ and $E$ on $AC$ and $AB$, respectively, and assume that they do not coincide with any of the vertices $A$, $B$, $C$. If the segments $BD$ and $CE$ intersect at $F$, consider areas $w$, $x$, $y$, $z$ of the quadrilateral $AEFD$ and the triangles $BEF$, $BFC$, $CDF$, respectively. [list=a] [*] Prove that $y^2 > xz$. [*] Determine $w$ in terms of $x$, $y$, $z$. [/list] [asy] import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(12); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -2.8465032978885407, xmax = 9.445649196374966, ymin = -1.7618066305534972, ymax = 4.389732795464592; /* image dimensions */ draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282)--cycle, linewidth(0.5)); /* draw figures */ draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117), linewidth(0.5)); draw((-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282), linewidth(0.5)); draw((5.672613975760373,-0.636902634996282)--(3.8295013012181283,2.816337276198864), linewidth(0.5)); draw((-0.7368327629589799,-0.5920813291311117)--(4.569287648059735,1.430279997142299), linewidth(0.5)); draw((5.672613975760373,-0.636902634996282)--(1.8844000180622977,1.3644681598392678), linewidth(0.5)); label("$y$",(2.74779188172294,0.23771684184669772),SE*labelscalefactor); label("$w$",(3.2941097703568736,1.8657441499758196),SE*labelscalefactor); label("$x$",(1.6660824622277512,1.0025618859342047),SE*labelscalefactor); label("$z$",(4.288408327670633,0.8168138037986672),SE*labelscalefactor); /* dots and labels */ dot((3.8295013012181283,2.816337276198864),dotstyle); label("$A$", (3.8732067323088435,2.925600853925651), NE * labelscalefactor); dot((-0.7368327629589799,-0.5920813291311117),dotstyle); label("$B$", (-1.1,-0.7565817154670613), NE * labelscalefactor); dot((5.672613975760373,-0.636902634996282),dotstyle); label("$C$", (5.763466626982254,-0.7784344310124186), NE * labelscalefactor); dot((4.569287648059735,1.430279997142299),dotstyle); label("$D$", (4.692683565259744,1.5051743434774234), NE * labelscalefactor); dot((1.8844000180622977,1.3644681598392678),dotstyle); label("$E$", (1.775346039954538,1.4942479857047448), NE * labelscalefactor); dot((2.937230516274804,0.8082418657164665),linewidth(4.pt) + dotstyle); label("$F$", (2.889834532767763,0.954), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]