be a triangle, 
its orthocenter, 
its circumradius. Let 
be the reflection of the point 
be the reflection of the point 
across the line 
,
be the reflection of the point 
across the line 
.
.
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
be a real number and let 
be arbitrary real numbers in 
.
and 
are then defined recursively by
.
for all sufficiently large 
is a prime number
Y by Amir Hossein, MasterYi, IcosahedralDice, Ninja-Girl, Destroyer74, nguyendangkhoa17112003, sholly, amar_04, AmirKhusrau, Cygnet, justJen, Lamboreghini, Stuart111, Adventure10, Mango247
To make the exalted atmosphere in geometry, i will establish the geometric marathon topic with the rule like 
: 
As the name says only geometry problems are allowed.
Only post about solution to given problem(ofc your solution to your problem doesn't count in table...) and something about given topic are allowed ( you may give the hints )
When you post solution of 
-th problem you should post 
-th problem(with number next to it...), otherwise you'll get 
point(s) and moderator will post next one(problem)...
After posting solution one should copy previous result table and add one point to himself or if it's first solved problem for he/she, then add yourself to table like This...
After every 
problems I will sort the table(so that one with most point's is on top).
If the -th problem isn't solved for next 
days from the time it was posted then the poster should post new -th problem. But if someone solve first(version) of -th problem it also counts...
Anyone who doesn't follow rules isn't welcome here and his posts will be deleted(from the moment he start doing that...)
So , let's start with problem:
Problem 1:
, circumcenter 
, 
is symetric to 
through 
. The perpendicular bisector of 
intersect 
at 
.
Prove that:
: 
As the name says only geometry problems are allowed.
Only post about solution to given problem(ofc your solution to your problem doesn't count in table...) and something about given topic are allowed ( you may give the hints )
When you post solution of 
-th problem you should post 
-th problem(with number next to it...), otherwise you'll get 
point(s) and moderator will post next one(problem)...
After posting solution one should copy previous result table and add one point to himself or if it's first solved problem for he/she, then add yourself to table like This...
After every 
problems I will sort the table(so that one with most point's is on top).
If the -th problem isn't solved for next 
days from the time it was posted then the poster should post new -th problem. But if someone solve first(version) of -th problem it also counts...
Anyone who doesn't follow rules isn't welcome here and his posts will be deleted(from the moment he start doing that...)So , let's start with problem:
Problem 1:
, circumcenter 
, 
is symetric to 
through 
. The perpendicular bisector of 
intersect 
at 
.Prove that:
This post has been edited 2 times. Last edited by brokendiamond, Aug 3, 2018, 4:59 PM
because
from symmetry w. r. t. BC. Q. E. D
X is foot of perpendicular from 
to 
Prove that 
=
is harmonic four. Then the lines 
,
,
,
also define the harmonic four. As the lines 
.
,
be the feet of perpindiculars from 
and 
be the tangent points of the inscribed circle of 
the middle of the side 
,
and 
a
meet 
be the feuerbach hyperbola ,now since 
are cevian triangles of 
are the poles of 
so if 
then 
is the pole of 
be the infinite point on the line 
we have that 
are collinear so by La-haire theorem we have 
are concurrent 
is the isogonal conjugate of 
are collinear.
,
cuts 
at 
intersects 
at 
is the bisector of 
( collection)
.
,
is the reflection of 
.
cuts 
.
cuts 
at 
intersects 
.
are concurrent
is a triangle satisfying 
.
satisfying 
.
.
.
.
and we are done
.
which has a circumcircle,let 
be the orthocenter of 
respectively.Lines 
pass through 
.
.
are concurrent.
.
,
.
satisfying 
,
satisfying 
,
satisfying 
.
.
,
be the circumcircle of 
.
.
are symmetry to line 
.
are concycle.
.
.
,
.
are tangent to 
.
.
is tangent to 
.
.
are collinear.
are tangent to 
