In △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.
[*] Prove that y2>xz.
[*] Determine w in terms of x, y, z.
[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] geometryratiosladder theoreminequalities