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KoMaL A Problems
KoMaL A Problems 2017/2018
A. 719
A. 719
Part of
KoMaL A Problems 2017/2018
Problems
(1)
HM is parallel to OI
Source: KöMaL A. 719
4/13/2018
Let
A
B
C
ABC
A
BC
be a scalene triangle with circumcenter
O
O
O
and incenter
I
I
I
. The
A
A
A
-excircle,
B
B
B
-excircle, and
C
C
C
-excircle of triangle
A
B
C
ABC
A
BC
touch
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
at points
A
1
A_1
A
1
,
B
1
B_1
B
1
, and
C
1
C_1
C
1
, respectively. Let
P
P
P
be the orthocenter of
A
B
1
C
1
AB_1C_1
A
B
1
C
1
and
H
H
H
be the orthocenter of
A
B
C
ABC
A
BC
. Show that if
M
M
M
is the midpoint of
P
A
1
PA_1
P
A
1
, then lines
H
M
HM
H
M
and
O
I
OI
O
I
are parallel.Michael Ren
geometry