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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2019 Mexico National Olympiad
6
6
Part of
2019 Mexico National Olympiad
Problems
(1)
Another orthocenter problem <3
Source: Mexico National Olympiad 2019 P6
11/12/2019
Let
A
B
C
ABC
A
BC
be a triangle such that
∠
B
A
C
=
4
5
∘
\angle BAC = 45^{\circ}
∠
B
A
C
=
4
5
∘
. Let
H
,
O
H,O
H
,
O
be the orthocenter and circumcenter of
A
B
C
ABC
A
BC
, respectively. Let
ω
\omega
ω
be the circumcircle of
A
B
C
ABC
A
BC
and
P
P
P
the point on
ω
\omega
ω
such that the circumcircle of
P
B
H
PBH
PB
H
is tangent to
B
C
BC
BC
. Let
X
X
X
and
Y
Y
Y
be the circumcenters of
P
H
B
PHB
P
H
B
and
P
H
C
PHC
P
H
C
respectively. Let
O
1
,
O
2
O_1,O_2
O
1
,
O
2
be the circumcenters of
P
X
O
PXO
PXO
and
P
Y
O
PYO
P
Y
O
respectively. Prove that
O
1
O_1
O
1
and
O
2
O_2
O
2
lie on
A
B
AB
A
B
and
A
C
AC
A
C
, respectively.
geometry
circumcircle