MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2017 Korea Junior Math Olympiad
6
6
Part of
2017 Korea Junior Math Olympiad
Problems
(1)
Circumcircle of medial triangle
Source: KJMO 2017 p6
7/26/2019
Let triangle
A
B
C
ABC
A
BC
be an acute scalene triangle, and denote
D
,
E
,
F
D,E,F
D
,
E
,
F
by the midpoints of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively. Let the circumcircle of
D
E
F
DEF
D
EF
be
O
1
O_1
O
1
, and its center be
N
N
N
. Let the circumcircle of
B
C
N
BCN
BCN
be
O
2
O_2
O
2
.
O
1
O_1
O
1
and
O
2
O_2
O
2
meet at two points
P
,
Q
P, Q
P
,
Q
.
O
2
O_2
O
2
meets
A
B
AB
A
B
at point
K
(
≠
B
)
K(\neq B)
K
(
=
B
)
and meets
A
C
AC
A
C
at point
L
(
≠
C
)
L(\neq C)
L
(
=
C
)
. Show that the three lines
E
F
,
P
Q
,
K
L
EF,PQ,KL
EF
,
PQ
,
K
L
are concurrent.
geometry
circumcircle
KJMO