,
be the projection from 
onto 
.
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
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 
and 
respectively such that 
to be a point such that it lies on both the circles 
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
A triangle 
is inscribed in a circle 
.Let 
the centroid of 
.
We draw the altitudes
of the given triangle .Rays 
and 
meet (C) at 
and 
.Prove that points 
are concyclic.
is inscribed in a circle 
.Let 
the centroid of 
.We draw the altitudes
of the given triangle .Rays 
and 
meet (C) at 
and 
.Prove that points 
are concyclic.This post has been edited 1 time. Last edited by stergiu, Apr 1, 2019, 4:21 AM
Reason: change of barycenter to centroid
Reason: change of barycenter to centroid
This is a beautiful problem
be the medial triangle WRT 
be reflection of 
,
bisects 
,
,
is a rectangle 
Hence 
concur at, suppose, 
cyclic
and use the fact that the orthic axis WRT 
and hence suppose orthic axis 
at 
passes through 
with 
.
.
be the midpoints of 
.
conclic. 
and so through Radical axis theorem 
all lie on a circle.
or 
for the second time.
.
is a cyclic quadrilateral, where 
is the midpoint of side 
and 
.
as 
lie on the Nine Point Circle of 
.
.
are concyclic. 
.
such that 
.
are collinear.
.
Since 
,
are concurrent at point 
.
This implies that quadrilateral 
is cyclic as desired.
be the second intersection of 
with 
.
so 
is cyclic.
is concyclic along the nine-point circle. Let 
and 
.
It can easily be checked that such a point is unique (say, by setting variables for the side lengths 
and solving for 
,
)
