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2000 Moldova National Olympiad
Problem 8
geo ineq with inradii
geo ineq with inradii
Source: Moldova 2000 Grade 12 P8
April 28, 2021
geometry
geometric inequality
Problem Statement
A circle with radius
r
r
r
touches the sides
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
at
E
,
F
,
G
,
H
E,F,G,H
E
,
F
,
G
,
H
, respectively. The inradii of the triangles
E
B
F
,
F
C
G
,
G
D
H
,
H
A
E
EBF,FCG,GDH,HAE
EBF
,
FCG
,
G
DH
,
H
A
E
are equal to
r
1
,
r
2
,
r
3
,
r
4
r_1,r_2,r_3,r_4
r
1
,
r
2
,
r
3
,
r
4
. Prove that
r
1
+
r
2
+
r
3
+
r
4
≥
2
(
2
−
2
)
r
.
r_1+r_2+r_3+r_4\ge2\left(2-\sqrt2\right)r.
r
1
+
r
2
+
r
3
+
r
4
≥
2
(
2
−
2
)
r
.
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