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China National Olympiad
1989 China National Olympiad
4
4
Part of
1989 China National Olympiad
Problems
(1)
China Mathematical Olympiad 1989 problem4
Source: China Mathematical Olympiad 1989 problem4
10/28/2013
Given a triangle
A
B
C
ABC
A
BC
, points
D
,
E
,
F
D,E,F
D
,
E
,
F
lie on sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively. Moreover, the radii of incircles of
△
A
E
F
,
△
B
F
D
,
△
C
D
E
\triangle AEF, \triangle BFD, \triangle CDE
△
A
EF
,
△
BF
D
,
△
C
D
E
are equal to
r
r
r
. Denote by
r
0
r_0
r
0
and
R
R
R
the radii of incircles of
△
D
E
F
\triangle DEF
△
D
EF
and
△
A
B
C
\triangle ABC
△
A
BC
respectively. Prove that
r
+
r
0
=
R
r+r_0=R
r
+
r
0
=
R
.
geometry
China