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1999 Turkey MO (2nd round)
5
Turkey NMO 1999, P-5, nice geometric inequality on altitudes
Turkey NMO 1999, P-5, nice geometric inequality on altitudes
Source:
December 23, 2010
inequalities
geometry
circumcircle
trigonometry
geometry proposed
Problem Statement
In an acute triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with circumradius
R
R
R
, altitudes
A
D
‾
,
B
E
‾
,
C
F
‾
\overline{AD},\overline{BE},\overline{CF}
A
D
,
BE
,
CF
have lengths
h
1
,
h
2
,
h
3
{{h}_{1}},{{h}_{2}},{{h}_{3}}
h
1
,
h
2
,
h
3
, respectively. If
t
1
,
t
2
,
t
3
{{t}_{1}},{{t}_{2}},{{t}_{3}}
t
1
,
t
2
,
t
3
are lengths of the tangents from
A
,
B
,
C
A,B,C
A
,
B
,
C
, respectively, to the circumcircle of triangle
△
D
E
F
\vartriangle DEF
△
D
EF
, prove that
∑
i
=
1
3
(
t
i
h
i
)
2
≤
3
2
R
\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R
i
=
1
∑
3
(
h
i
t
i
)
2
≤
2
3
R
.
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