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Colombia National Olympiad
2020 Colombia National Olympiad
5
5
Part of
2020 Colombia National Olympiad
Problems
(1)
Geometry from Colombia
Source: Colombia 2020
2/16/2022
Given an acute-angled triangle
A
B
C
ABC
A
BC
with
D
D
D
is the foot of the altitude from
A
.
A.
A
.
The perpendicular lines to
B
C
BC
BC
through
B
B
B
and
C
C
C
intersect the altitudes from
C
C
C
and
B
B
B
at points
M
M
M
and
N
N
N
, respectively. Show that
A
D
AD
A
D
=
=
=
B
C
BC
BC
if and only if
A
,
M
,
N
A,M,N
A
,
M
,
N
and
D
D
D
lie on the same circle.
Columbia
geometry