are positive real numbers, 
then
be positive real numbers with 
.
and 
Prove that
Y by Mango247
Three line segments divide a triangle into five triangles. The area of these triangles is called 
yand 
, as in the figure.
(a) Prove that
.
(b) Prove that the area of the great triangle is at most
yand 
, as in the figure.(a) Prove that
.(b) Prove that the area of the great triangle is at most
This post has been edited 2 times. Last edited by parmenides51, Dec 23, 2022, 7:01 PM
,
and
we get that 
.![[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]](http://latex.artofproblemsolving.com/f/1/a/f1aacb2c4e73e9dace5a6a44671de5d3a46454fe.png)
,
.
![[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]](http://latex.artofproblemsolving.com/9/2/7/927a38a84955cc4b21d1b125af83fcc2c8265cbd.png)
