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Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2024 Philippine Math Olympiad
P3
P3
Part of
2024 Philippine Math Olympiad
Problems
(1)
Geometry with orthocenter I
Source: Philippine Mathematical Olympiad 2024 P3
2/24/2024
Given triangle
A
B
C
ABC
A
BC
with orthocenter
H
H
H
, the lines through
B
B
B
and
C
C
C
perpendicular to
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, intersect line
A
H
AH
A
H
at
X
X
X
and
Y
Y
Y
, respectively. The circle with diameter
X
Y
XY
X
Y
intersects lines
B
X
BX
BX
and
C
Y
CY
C
Y
a second time at
K
K
K
and
L
L
L
, respectively. Prove that points
H
,
K
H, K
H
,
K
and
L
L
L
are collinear.
geometry
orthocenter