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239 Open Math Olympiad
2018 239 Open Mathematical Olympiad
8-9.2
8-9.2
Part of
2018 239 Open Mathematical Olympiad
Problems
(1)
Right-angled triangle
Source: 239 Open MO 2018, Junior League, Problem 2
4/4/2023
On the hypotenuse
A
B
AB
A
B
of a right-angled triangle
A
B
C
ABC
A
BC
, point
R
R
R
is chosen, on the cathetus
B
C
BC
BC
a point
T
T
T
, and on the segment
A
T
AT
A
T
a point
S
S
S
are chosen in such a way that the angles
∠
A
R
T
\angle ART
∠
A
RT
and
∠
A
S
C
\angle ASC
∠
A
SC
are right angles. Points
M
M
M
and
N
N
N
are the midpoints of the segments
C
B
CB
CB
and
C
R
CR
CR
, respectively. Prove that points
M
M
M
,
T
T
T
,
S
S
S
, and
N
N
N
lie on the same circle.Proposed by S. Berlov
geometry