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Mediterranean Mathematics Olympiad
2006 Mediterranean Mathematics Olympiad
2
Area Inequality
Area Inequality
Source: Mediterranean MO 2006
October 30, 2010
geometry
inequalities
parallelogram
geometry proposed
Problem Statement
Let
P
P
P
be a point inside a triangle
A
B
C
ABC
A
BC
, and
A
1
B
2
,
B
1
C
2
,
C
1
A
2
A_1B_2,B_1C_2,C_1A_2
A
1
B
2
,
B
1
C
2
,
C
1
A
2
be segments passing through
P
P
P
and parallel to
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
respectively, where points
A
1
,
A
2
A_1, A_2
A
1
,
A
2
lie on
B
C
,
B
1
,
B
2
BC, B_1, B_2
BC
,
B
1
,
B
2
on
C
A
CA
C
A
, and
C
1
,
C
2
C_1,C_2
C
1
,
C
2
on
A
B
AB
A
B
. Prove that
Area
(
A
1
A
2
B
1
B
2
C
1
C
2
)
≥
1
2
Area
(
A
B
C
)
\text{Area}(A_1A_2B_1B_2C_1C_2) \ge \frac{1}{2}\text{Area}(ABC)
Area
(
A
1
A
2
B
1
B
2
C
1
C
2
)
≥
2
1
Area
(
A
BC
)
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