in any traingle abc the minimum value of r1+r2+r3/r is
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Least value of
r
1
+
r
2
+
r
3
r
is
9
Explanation:
For a
Δ
A
B
C
, exradii
r
1
=
Δ
s
−
a
,
r
2
=
Δ
s
−
b
,
r
3
=
Δ
s
−
c
and inradius
r
=
Δ
s
, where
s
is semiperimeter of the triangle.
r
+
r
2
+
r
3
−
r
=
Δ
s
−
a
+
Δ
s
−
b
+
Δ
s
−
c
−
Δ
s
=
Δ
s
(
s
−
b
)
(
s
−
c
)
+
s
(
s
−
a
)
(
s
−
c
)
+
s
(
s
−
a
)
(
s
−
b
)
−
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
Δ
2
=
s
(
s
−
c
)
(
s
−
b
+
s
−
a
)
+
(
s
−
a
)
(
s
−
b
)
(
s
−
s
+
c
)
Δ
=
c
s
(
s
−
c
)
+
c
(
s
−
a
)
(
s
−
b
)
Δ
=
c
(
s
2
−
s
c
+
s
2
−
a
s
−
b
s
+
a
b
)
Δ
=
c
(
2
s
2
−
s
⋅
2
s
+
a
b
)
Δ
=
a
b
c
Δ
=
4
R
Hence
r
1
+
r
2
+
r
3
r
=
4
R
r
+
1
(A)
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