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3
an elegant identity in elements of a triangle
an elegant identity in elements of a triangle
Source: Moldova 2007 IMO-BMO TST III problem 3
March 24, 2007
geometry
inradius
circumcircle
trigonometry
geometry proposed
Problem Statement
Consider a triangle
A
B
C
ABC
A
BC
, with corresponding sides
a
,
b
,
c
a,b,c
a
,
b
,
c
, inradius
r
r
r
and circumradius
R
R
R
. If
r
A
,
r
B
,
r
C
r_{A}, r_{B}, r_{C}
r
A
,
r
B
,
r
C
are the radii of the respective excircles of the triangle, show that
a
2
(
2
r
A
−
r
r
B
r
C
)
+
b
2
(
2
r
B
−
r
r
A
r
C
)
+
c
2
(
2
r
C
−
r
r
A
r
B
)
=
4
(
R
+
3
r
)
a^{2}\left(\frac 2{r_{A}}-\frac{r}{r_{B}r_{C}}\right)+b^{2}\left(\frac 2{r_{B}}-\frac{r}{r_{A}r_{C}}\right)+c^{2}\left(\frac 2{r_{C}}-\frac{r}{r_{A}r_{B}}\right)=4(R+3r)
a
2
(
r
A
2
−
r
B
r
C
r
)
+
b
2
(
r
B
2
−
r
A
r
C
r
)
+
c
2
(
r
C
2
−
r
A
r
B
r
)
=
4
(
R
+
3
r
)
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