be an acute triangle with 
,
be the projection from 
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
intersect 
.
.
sastifying 
then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is the incentre and point 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
respectively such that 
and 
are perpendicular.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
.
.
to meet 
meets the midpoint of 
,
.
with side length of 
.
.
as 
.
Y by Adventure10
Let 
be two externally tangent circles at point 
. A line 
is tangent to 
in point 
and intersects 
in points 
such that 
is closer to than 
is. The line 
intersects the perpendicular to in at point 
, the line 
intersects 
in 
.
Prove that
passes through the midpoint of 
.
be two externally tangent circles at point 
. A line 
is tangent to 
in point 
and intersects 
in points 
such that 
is closer to than 
is. The line 
intersects the perpendicular to in at point 
, the line 
intersects 
in 
.Prove that
passes through the midpoint of 
.This post has been edited 2 times. Last edited by Tsarik, Feb 19, 2016, 9:07 AM
the reflection of 
and 
intersects the perpendicular to 
and let 
and 
is perpendicular bisector of 
and let 
and 
are internal bisector and external bisector of 
but also that 
are collinear 
and 
since 
we get 
