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Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
1995 South africa National Olympiad
4
4
Part of
1995 South africa National Olympiad
Problems
(1)
Ratio between areas of triangles defined by tangent circles
Source: South African MO 1995 Q4
5/27/2012
Three circles, with radii
p
p
p
,
q
q
q
and
r
r
r
and centres
A
A
A
,
B
B
B
and
C
C
C
respectively, touch one another externally at points
D
D
D
,
E
E
E
and
F
F
F
. Prove that the ratio of the areas of
△
D
E
F
\triangle DEF
△
D
EF
and
△
A
B
C
\triangle ABC
△
A
BC
equals
2
p
q
r
(
p
+
q
)
(
q
+
r
)
(
r
+
p
)
.
\frac{2pqr}{(p+q)(q+r)(r+p)}.
(
p
+
q
)
(
q
+
r
)
(
r
+
p
)
2
pq
r
.
ratio
geometry
geometry unsolved