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2009 Ukraine Team Selection Test
12
sum h_a^{2}/(a ^2-CH_a ^2) >= 3, altitudes
sum h_a^{2}/(a ^2-CH_a ^2) >= 3, altitudes
Source: Ukraine TST 2009 p12
May 4, 2020
altitudes
geometric inequality
geometry
Problem Statement
Denote an acute-angle
△
A
B
C
\vartriangle ABC
△
A
BC
with sides
a
,
b
,
c
a, b, c
a
,
b
,
c
respectively by
H
a
,
H
b
,
H
c
{{H}_{a}}, {{H}_{b}}, {{H}_{c}}
H
a
,
H
b
,
H
c
the feet of altitudes
h
a
,
h
b
,
h
c
{{h}_{a}}, {{h}_{b}}, {{h}_{c}}
h
a
,
h
b
,
h
c
. Prove the inequality:
h
a
2
a
2
−
C
H
a
2
+
h
b
2
b
2
−
A
H
b
2
+
h
c
2
c
2
−
B
H
c
2
≥
3
\frac {h_ {a} ^{2}} {{{a} ^{2}} - CH_ {a} ^{2}} + \frac{h_{b} ^{2}} {{{ b}^{2}} - AH_{b} ^{2}} + \frac{h_{c}^{2}}{{{c}^{2}} - BH_{c}^{2}} \ge 3
a
2
−
C
H
a
2
h
a
2
+
b
2
−
A
H
b
2
h
b
2
+
c
2
−
B
H
c
2
h
c
2
≥
3
(Dmitry Petrovsky)
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