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Korea Junior Mathematics Olympiad
2019 Korea Junior Math Olympiad.
2
Prove angle ADB=3angleBAC if AE=CD
Prove angle ADB=3angleBAC if AE=CD
Source: KJMO 2019 p2
January 8, 2021
geometry
KJMO
Problem Statement
In an acute triangle
A
B
C
ABC
A
BC
, point
D
D
D
is on the segment
A
C
AC
A
C
such that
A
D
‾
=
B
C
‾
\overline{AD}=\overline{BC}
A
D
=
BC
and
A
C
‾
2
−
A
D
‾
2
=
A
C
‾
⋅
A
D
‾
\overline{AC}^2-\overline{AD}^2=\overline{AC}\cdot\overline{AD}
A
C
2
−
A
D
2
=
A
C
⋅
A
D
. The line that is parallel to the bisector of
∠
A
C
B
\angle{ACB}
∠
A
CB
and passes the point
D
D
D
meets the segment
A
B
AB
A
B
at point
E
E
E
. Prove, if
A
E
‾
=
C
D
‾
\overline{AE}=\overline{CD}
A
E
=
C
D
,
∠
A
D
B
=
3
∠
B
A
C
\angle{ADB}=3\angle{BAC}
∠
A
D
B
=
3∠
B
A
C
.
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