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 
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, Mango247
On a line containing the altitude 
of the triangle 
, a point 
is taken that is different from 
and 
such that the lines 
and 
intersect the lines 
and 
at the points 
and 
respectively. Perpendiculars 
, 
, 
, 
on lines 
and 
are pass through points 
and . Prove that the lines 
, 
and 
intersect at the same point.
a) Solve the problem when is the intersection point of the altitudes of the triangle .
b) Solve the problem for an arbitrary point.
of the triangle 
, a point 
is taken that is different from 
and 
such that the lines 
and 
intersect the lines 
and 
at the points 
and 
respectively. Perpendiculars 
, 
, 
, 
on lines 
and 
are pass through points 
and . Prove that the lines 
, 
and 
intersect at the same point.a) Solve the problem when
is the intersection point of the altitudes of the triangle .b) Solve the problem for an arbitrary point
.This post has been edited 1 time. Last edited by parmenides51, Sep 28, 2018, 6:01 AM
Reason: name typo
Reason: name typo
