and 
.
and 
.
Equality holds when 
or 
be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
,
intersect 
at 
.
and 
are either parallel or coincide.
.
Where 
Y by Adventure10, Mango247
In the triangle 
, side 
is greater than side 
. On the extension of the side beyond the point 
, noted the point 
so that 
. Let 
be the angle bisector of , 
be the midpoint of the side 
, а 
be a point on the side such that 
. Prove that 
.
, side 
is greater than side 
. On the extension of the side beyond the point 
, noted the point 
so that 
. Let 
be the angle bisector of , 
be the midpoint of the side 
, а 
be a point on the side such that 
. Prove that 
.
be on 
such that 
,
.
.
implies 
,
,
.
through 
and 
through line 
,
.
lies on ![\[\triangle CLN \sim \triangle CC'A \implies LN \parallel A'C' \implies LN \parallel A'S \implies \triangle BLN \sim \triangle BSA'\]](http://latex.artofproblemsolving.com/c/b/5/cb501dfcc2416282323792ab2870174ed24d8da1.png)
This together with 
yields![\[\frac{LN}{A'S} = \frac{BL}{BS} = \frac{AM}{AS} = \frac{\frac{1}{2}AC}{AS} = \frac{\frac{1}{2}AC}{A'S}\]](http://latex.artofproblemsolving.com/a/8/f/a8fdfda9650dea6953096f22ffbe2b91f032fd74.png)
Thus 
be the reflection of 
then 
and since 
i
l
