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2019 JBMO Shortlist
G7
G7
Part of
2019 JBMO Shortlist
Problems
(1)
JBMO Shortlist 2019 G7
Source:
9/12/2020
Let
A
B
C
ABC
A
BC
be a right-angled triangle with
∠
A
=
9
0
∘
\angle A = 90^{\circ}
∠
A
=
9
0
∘
. Let
K
K
K
be the midpoint of
B
C
BC
BC
, and let
A
K
L
M
AKLM
A
K
L
M
be a parallelogram with centre
C
C
C
. Let
T
T
T
be the intersection of the line
A
C
AC
A
C
and the perpendicular bisector of
B
M
BM
BM
. Let
ω
1
\omega_1
ω
1
be the circle with centre
C
C
C
and radius
C
A
CA
C
A
and let
ω
2
\omega_2
ω
2
be the circle with centre
T
T
T
and radius
T
B
TB
TB
. Prove that one of the points of intersection of
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
is on the line
L
M
LM
L
M
.Proposed by Greece
geometry