.
as 
,
and 
as 
.
.
are there such that 
and 
?
, where 
is a real number. Prove that for all positive 
then prove
Y by Adventure10
Let 
be a triangle and points 
on the sides 
and 
respectively in such a way that 
for 
. Segments 
and 
are cut in 
, segments 
and are cut at 
and segments and are cut at 
. Calculate the ratio of areas of the triangles 
.
be a triangle and points 
on the sides 
and 
respectively in such a way that 
for 
. Segments 
and 
are cut in 
, segments 
and are cut at 
and segments and are cut at 
. Calculate the ratio of areas of the triangles 
.This post has been edited 1 time. Last edited by parmenides51, Jul 23, 2021, 3:57 PM
Reason: title
Reason: title
Proof:
since 
is trivial)
for line 
Where 
So, 
![$$\frac {[BFC]}{[BRC]} = \frac {n^2 - n}{n^2-n+1} , \frac {[BRC]}{[ABC]} = \frac {1}{n}$$](http://latex.artofproblemsolving.com/8/6/6/86634cda30197595dcf9c57601a04531c3b2b460.png)
So,![$$\boxed{\frac {[BFC]}{[ABC]} = \frac {n-1}{n^2 - n +1}}$$](http://latex.artofproblemsolving.com/8/2/e/82e26c7c3995d7673a6def3bdfaa99c77f859c55.png)
Similarly, we get this for ![$[CDA] , [AEB]$](http://latex.artofproblemsolving.com/0/1/d/01d87c06243b89834e93f096e21cdb339fe26e0e.png)
![$$[ABC] = [DEF] + [BFC] + [CDA] + [AEB]$$](http://latex.artofproblemsolving.com/b/a/3/ba306d198bdb7b342a7c671d69630fa7294eb7c6.png)
