be an acute triangle with 
,
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 
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 
,
.
.
.
as 
.
Y by Vanuatu, Adventure10, Mango247
Let 
be a quadrilateral whose diagonals are not perpendicular and whose sides 
and 
are not parallel.Let 
be the intersection of its diagonals.Denote with 
and 
the orthocenters of triangles 
and 
respectively.If 
and 
are the midpoints of the segment lines and 
respectively.Prove that the lines 
and 
are parallel if and only if 
be a quadrilateral whose diagonals are not perpendicular and whose sides 
and 
are not parallel.Let 
be the intersection of its diagonals.Denote with 
and 
the orthocenters of triangles 
and 
respectively.If 
and 
are the midpoints of the segment lines and 
respectively.Prove that the lines 
and 
are parallel if and only if 
This post has been edited 1 time. Last edited by Orkhan-Ashraf_2002, Nov 7, 2016, 3:38 PM
.
are midpoints of 
then 
.
.
is always a parallelogram, 
is a rhombus, which happens iff 
.
