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Inequality on triangle with altitudes [Taiwan 2014 Quizzes]
Inequality on triangle with altitudes [Taiwan 2014 Quizzes]
Source:
July 18, 2014
inequalities
geometry
area of a triangle
Heron's formula
Problem Statement
A triangle has side lengths
a
a
a
,
b
b
b
,
c
c
c
, and the altitudes have lengths
h
a
h_a
h
a
,
h
b
h_b
h
b
,
h
c
h_c
h
c
. Prove that
(
a
h
a
)
2
+
(
b
h
b
)
2
+
(
c
h
c
)
2
≥
4.
\left( \frac{a}{h_a} \right)^2 + \left( \frac{b}{h_b} \right)^2 + \left( \frac{c}{h_c} \right)^2 \ge 4.
(
h
a
a
)
2
+
(
h
b
b
)
2
+
(
h
c
c
)
2
≥
4.
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