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Problems
Contests
National and Regional Contests
Hong Kong Contests
Hong Kong National Olympiad
1999 Hong kong National Olympiad
2
2
Part of
1999 Hong kong National Olympiad
Problems
(1)
Angle AIO is less than 90 degrees iff 2a =<b+c
Source:
11/7/2010
Let
I
I
I
be the incentre and
O
O
O
the circumcentre of a non-equilateral triangle
A
B
C
ABC
A
BC
. Prove that
∠
A
I
O
≤
9
0
∘
\angle AIO \le 90^{\circ}
∠
A
I
O
≤
9
0
∘
if and only if
2
B
C
≤
A
B
+
A
C
2BC\le AB+AC
2
BC
≤
A
B
+
A
C
.
geometry
incenter
trigonometry
geometry unsolved