We know that $\sin{(\alpha)}=\sin{(180^{\circ}-\alpha)}$, thus since $$uv=(a\cdot b\cdot \sin{(\alpha)}\cdot \frac{1}{2})(c\cdot d \cdot \sin{(\alpha)}\cdot \frac{1}{2})$$and $$yz=(a\cdot c\cdot \sin{(180^{\circ}-\alpha)}\cdot \frac{1}{2})(b\cdot d \cdot \sin{(180^{\circ}-\alpha)}\cdot \frac{1}{2}),$$we get that $uv=yz$. [asy][asy] import graph; size(6cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -0.9792804048965501, xmax = 7.579182926158856, ymin = -9.636552989146839, ymax = 8.046742569872043; /* image dimensions */ /* draw figures */ draw((2.51,1.72)--(6.07,1.66), linewidth(1)); draw((4.817016330757588,4.343276348283279)--(3.2551791907514454,4.038335260115607), linewidth(1)); draw((3.2551791907514454,4.038335260115607)--(3.95,6.2), linewidth(1)); draw((3.2551791907514454,4.038335260115607)--(2.51,1.72), linewidth(1)); draw((4.817016330757588,4.343276348283279)--(3.95,6.2), linewidth(1)); draw((4.817016330757588,4.343276348283279)--(6.07,1.66), linewidth(1)); draw((3.255179190751446,4.038335260115606)--(3.997352762692937,3.4112482464503873), linewidth(1)); draw((3.997352762692937,3.4112482464503873)--(4.817016330757587,4.343276348283279), linewidth(1)); draw((3.997352762692937,3.4112482464503873)--(2.51,1.72), linewidth(1)); draw((3.997352762692937,3.4112482464503873)--(6.07,1.66), linewidth(1)); /* dots and labels */ label("$a$", (3.45727595421673,3.5472563333245346), NE * labelscalefactor); label("$c$", (4.590013748032887,3.7203134962686693), NE * labelscalefactor); label("$b$", (3.063964220252787,2.7449003960380907), NE * labelscalefactor); label("$d$", (4.8574657271283685,2.3515886620741475), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/asy]

We need to prove that $S\le \frac{xz}{y}$, where we denote the area of the big triangle as $S$. By Menelaus's theorem, $\frac{c}{b}\cdot \frac{e}{f}\cdot \frac{(g+h)}{h}=1$ $\frac{a}{d}\cdot \frac{h}{g}\cdot \frac{(e+f)}{e}=1$ $\frac{ac}{bd}\cdot \frac{e}{f}\cdot \frac{(g+h)}{h}\cdot \frac{h}{g}\cdot \frac{(e+f)}{e}=1$ $\frac{y}{z}\cdot \frac{(g+h)}{g}\cdot \frac{(e+f)}{f}=1$ $\frac{y}{z}\cdot \frac{S}{x}=1\implies S=\frac{xz}{y}$ [asy][asy] import graph; size(6cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -0.9792804048965501, xmax = 7.579182926158856, ymin = -9.636552989146839, ymax = 8.046742569872043; /* image dimensions */ /* draw figures */ draw((2.51,1.72)--(6.07,1.66), linewidth(1)); draw((4.817016330757588,4.343276348283279)--(3.2551791907514454,4.038335260115607), linewidth(1)); draw((3.2551791907514454,4.038335260115607)--(3.95,6.2), linewidth(1)); draw((3.2551791907514454,4.038335260115607)--(2.51,1.72), linewidth(1)); draw((4.817016330757588,4.343276348283279)--(3.95,6.2), linewidth(1)); draw((4.817016330757588,4.343276348283279)--(6.07,1.66), linewidth(1)); draw((3.255179190751446,4.038335260115606)--(3.997352762692937,3.4112482464503873), linewidth(1)); draw((3.997352762692937,3.4112482464503873)--(4.817016330757587,4.343276348283279), linewidth(1)); draw((3.997352762692937,3.4112482464503873)--(2.51,1.72), linewidth(1)); draw((3.997352762692937,3.4112482464503873)--(6.07,1.66), linewidth(1)); /* dots and labels */ label("$f$", (3.4415434848581725,5.262095493407326), NE * labelscalefactor); label("$e$", (2.639187547571728,2.9651549670578987), NE * labelscalefactor); label("$g$", (4.464153993164425,5.340757840200115), NE * labelscalefactor); label("$h$", (5.203580053016638,2.9022250896236677), NE * labelscalefactor); label("$a$", (3.45727595421673,3.5472563333245346), NE * labelscalefactor); label("$c$", (4.590013748032887,3.7203134962686693), NE * labelscalefactor); label("$b$", (3.063964220252787,2.7449003960380907), NE * labelscalefactor); label("$d$", (4.8574657271283685,2.3515886620741475), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]