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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2023 Korea Junior Math Olympiad
2
2
Part of
2023 Korea Junior Math Olympiad
Problems
(1)
DFG = FCG
Source: KJMO 2023 P2
11/4/2023
Quadrilateral
A
B
C
D
(
A
D
‾
<
B
C
‾
)
ABCD (\overline{AD} < \overline{BC})
A
BC
D
(
A
D
<
BC
)
is inscribed in a circle, and
H
(
≠
A
,
B
)
H(\neq A, B)
H
(
=
A
,
B
)
is a point on segment
A
B
.
AB.
A
B
.
The circumcircle of triangle
B
C
H
BCH
BC
H
meets
B
D
BD
B
D
at
E
(
≠
B
)
E(\neq B)
E
(
=
B
)
and line
H
E
HE
H
E
meets
A
D
AD
A
D
at
F
F
F
. The circle passes through
C
C
C
and tangent to line
B
D
BD
B
D
at
E
E
E
meets
E
F
EF
EF
at
G
(
≠
E
)
.
G(\neq E).
G
(
=
E
)
.
Prove that
∠
D
F
G
=
∠
F
C
G
.
\angle DFG = \angle FCG.
∠
D
FG
=
∠
FCG
.
geometry
circumcircle