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National and Regional Contests
Mexico Contests
Mexico National Olympiad
2021 Mexico National Olympiad
2
2
Part of
2021 Mexico National Olympiad
Problems
(1)
midpoints equivalency and a tangent ?
Source: Mexico National Olympiad 2021 Problem 2
11/10/2021
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
C
B
>
9
0
∘
\angle ACB > 90^{\circ}
∠
A
CB
>
9
0
∘
, and let
D
D
D
be a point on
B
C
BC
BC
such that
A
D
AD
A
D
is perpendicular to
B
C
BC
BC
. Consider the circumference
Γ
\Gamma
Γ
with with diameter
B
C
BC
BC
. A line
ℓ
\ell
ℓ
passes through
D
D
D
and is tangent to
Γ
\Gamma
Γ
at
P
P
P
, cuts
A
C
AC
A
C
at
M
M
M
(such that
M
M
M
is in between
A
A
A
and
C
C
C
), and cuts the side
A
B
AB
A
B
at
N
N
N
. Prove that
M
M
M
is the midpoint of
D
P
DP
D
P
if and only if
N
N
N
is the midpoint of
A
B
AB
A
B
.
geometry