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International Contests
IMO Shortlist
1985 IMO Shortlist
21
21
Part of
1985 IMO Shortlist
Problems
(1)
Prove that BAM=CAX and AM/AX=cos(BAC)
Source:
8/29/2010
The tangents at
B
B
B
and
C
C
C
to the circumcircle of the acute-angled triangle
A
B
C
ABC
A
BC
meet at
X
X
X
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Prove that(a)
∠
B
A
M
=
∠
C
A
X
\angle BAM = \angle CAX
∠
B
A
M
=
∠
C
A
X
, and(b)
A
M
A
X
=
cos
∠
B
A
C
.
\frac{AM}{AX} = \cos\angle BAC.
A
X
A
M
=
cos
∠
B
A
C
.
geometry
circumcircle
trigonometry
Triangle
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