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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2021 Moldova Team Selection Test
3
3
Part of
2021 Moldova Team Selection Test
Problems
(1)
Show that the circumcenter of triangle MDN lies on the line BC
Source: Moldova TST 2021
9/19/2021
Acute triangle
A
B
C
ABC
A
BC
with
A
B
>
B
C
AB>BC
A
B
>
BC
is inscribed in circle
Ω
\Omega
Ω
. Points
D
D
D
and
E
E
E
, that lie on
(
B
C
)
(BC)
(
BC
)
and
(
A
B
)
(AB)
(
A
B
)
are the feet of altitudes from
A
A
A
and
C
C
C
in triangle
A
B
C
ABC
A
BC
, and
M
M
M
is the midpoint of the segment
D
E
DE
D
E
. Half-line
(
A
M
(AM
(
A
M
intersects the circle
Ω
\Omega
Ω
for the second time in
N
N
N
. Show that the circumcenter of triangle
M
D
N
MDN
M
D
N
lies on the line
B
C
BC
BC
.
geometry
circumcircle