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Baltic Way
2019 Baltic Way
14
14
Part of
2019 Baltic Way
Problems
(1)
Medians of right triangles form bisecting lines
Source: 2019 Baltic Way P14
11/18/2019
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
B
C
=
9
0
∘
\angle ABC = 90^{\circ}
∠
A
BC
=
9
0
∘
, and let
H
H
H
be the foot of the altitude from
B
B
B
. The points
M
M
M
and
N
N
N
are the midpoints of the segments
A
H
AH
A
H
and
C
H
CH
C
H
, respectively. Let
P
P
P
and
Q
Q
Q
be the second points of intersection of the circumcircle of the triangle
A
B
C
ABC
A
BC
with the lines
B
M
BM
BM
and
B
N
BN
BN
, respectively. The segments
A
Q
AQ
A
Q
and
C
P
CP
CP
intersect at the point
R
R
R
. Prove that the line
B
R
BR
BR
passes through the midpoint of the segment
M
N
MN
MN
.
geometry