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National and Regional Contests
China Contests
China Western Mathematical Olympiad
2012 China Western Mathematical Olympiad
4
Geometric inequality
Geometric inequality
Source: 2012 CWMO P4
September 30, 2012
inequalities
geometry
circumcircle
geometry proposed
Problem Statement
P
P
P
is a point in the
Δ
A
B
C
\Delta ABC
Δ
A
BC
,
ω
\omega
ω
is the circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
.
B
P
∩
ω
=
{
B
,
B
1
}
BP \cap \omega = \left\{ {B,{B_1}} \right\}
BP
∩
ω
=
{
B
,
B
1
}
,
C
P
∩
ω
=
{
C
,
C
1
}
CP \cap \omega = \left\{ {C,{C_1}} \right\}
CP
∩
ω
=
{
C
,
C
1
}
,
P
E
⊥
A
C
PE \bot AC
PE
⊥
A
C
，
P
F
⊥
A
B
PF \bot AB
PF
⊥
A
B
. The radius of the inscribed circle and circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
is
r
,
R
r,R
r
,
R
. Prove
E
F
B
1
C
1
⩾
r
R
\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}
B
1
C
1
EF
⩾
R
r
.
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