Y by wanwan4343, buratinogigle, mineiraojose, mhq, Gryphos, buzzychaoz, Scorpion.k48, Vanescralet, baopbc, aopser123, JasperL, AlastorMoody, parola, RAMUGAUSS, AmirKhusrau, Functional_equation, PIartist, Multily, Adventure10, bin_sherlo, MS_asdfgzxcvb
Some properties of Hagge circle 
Theorem 1:
Let
be the orthocenter of
.
Let
be the circumcevian triangle of
WRT
.
Let
be the reflection of
in
, respectively .
Then
are concyclic .
Proof:
Let
be the isogonal conjugate of
WRT
.
Let
be the circumcevian triangle of
WRT
.
Let
be the reflection of
in the midpoint
of
, respectively .
Let
be the circumcenter of
and
be the antipode of
in
.
Easy to see
is the reflection of
in the midpoint of
, respectively .
Since
are parallelogram ,
so
is a parallelogram
the midpoint of
is the projection of
on
.
Similarly we can get the midpoint of
is the projection of
on
,
so the midpoint of
lie on a circle with diameter
,
hence after doing homothety
we get
are concyclic .
Notice that
is the reflection of
in
, respectively ,
so we get
are concyclic . 
The circle in Theorem 1 is called the Hagge circle
of
WRT
.
____________________________________________________________
Theorem 2:
The center
of
is the reflection of
in the 9-point center
of
.
Proof:
Let
be the image of
under homothety
.
From the proof of Theorem 1 we get the reflection of
in
is the antipode of
in
,
so from
and
is a parallelogram ,
hence we get
is the reflection of
in the midpoint
of
. 
____________________________________________________________
Theorem 3:
Let
.
Then
.
Proof:
Lemma:
Let
be the isogonal conjugate of
.
Let
.
Then
.
Proof of the lemma:
Since
,
so we get
. ... 
Since
,
so we get
. ... 
From
. i.e. 
Back to the main proof:
Let
be the antipode of
in
.
From symmetry it suffices to prove
.
Since
and
,
so
is the anti-complement of
WRT
,
hence from
we get
,
so combine with the lemma
. 
____________________________________________________________
Theorem 4:
.
Proof:
Let
be the midpoint of
, respectively .
From the proof of Theorem 1 we get
is the projection of
on
, respectively .
Since
(Similarly
) ,
so we get
.
Since
,
so we get
(Similar discussion for
and
) ,
hence combine with Theorem 3
. 
____________________________________________________________
Some topics related to Hagge circle:
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=444395
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=320075
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=318484
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=50&t=329272

Theorem 1:
Let


Let



Let



Then

Proof:
Let



Let



Let




Let





Easy to see



Since

so





Similarly we can get the midpoint of



so the midpoint of


hence after doing homothety


Notice that



so we get


The circle in Theorem 1 is called the Hagge circle



____________________________________________________________
Theorem 2:
The center





Proof:
Let



From the proof of Theorem 1 we get the reflection of




so from


hence we get





____________________________________________________________
Theorem 3:
Let

Then

Proof:
Lemma:
Let


Let

Then

Proof of the lemma:
Since

so we get


Since

so we get


From


Back to the main proof:
Let



From symmetry it suffices to prove

Since


so



hence from


so combine with the lemma


____________________________________________________________
Theorem 4:

Proof:
Let


From the proof of Theorem 1 we get



Since


so we get

Since

so we get



hence combine with Theorem 3


____________________________________________________________
Some topics related to Hagge circle:
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=444395
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=320075
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=318484
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=50&t=329272