MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2023 Argentina National Olympiad
3
3
Part of
2023 Argentina National Olympiad
Problems
(1)
ARGENTINA MO National Level 3 P3
Source:
3/21/2024
Let
A
B
C
ABC
A
BC
be a triangle and
M
M
M
be the middle point of
B
C
BC
BC
. Let
Ω
\Omega
Ω
be the circumference such as
A
,
B
,
C
∈
Ω
A,B,C \in \Omega
A
,
B
,
C
∈
Ω
. Let
P
P
P
be the intersection of
Ω
\Omega
Ω
and
A
M
AM
A
M
.
A
F
AF
A
F
is a hight of the triangle, with
F
∈
B
C
F\in BC
F
∈
BC
, and
H
H
H
the orthocenter.Additionally the intersections of
M
H
MH
M
H
and
P
F
PF
PF
with
Ω
\Omega
Ω
are
K
K
K
and
T
T
T
respectibly. Demonstrate that the circumscribed circumference of the traingle
K
T
F
KTF
K
TF
is tangent with
B
C
BC
BC
.
geometry
tangent