MathDB
Problems
Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2003 South africa National Olympiad
6
6
Part of
2003 South africa National Olympiad
Problems
(1)
This is insane
Source: South Africa 2003
10/14/2005
In
Δ
A
B
C
\Delta ABC
Δ
A
BC
, the sum of the sides is
2
s
2s
2
s
and the radius of the incircle is
r
r
r
. Three semicircles with diameters
A
B
AB
A
B
,
B
C
BC
BC
and
C
A
CA
C
A
are drawn on the outside of
A
B
C
ABC
A
BC
. A circle with radius
t
t
t
touches all three semicircles. Prove that
s
2
<
t
≤
s
2
+
(
1
−
3
2
)
r
.
\frac{s}{2} < t \leq \frac{s}{2} + \left(1 - \frac{\sqrt{3}}{2}\right)r.
2
s
<
t
≤
2
s
+
(
1
−
2
3
)
r
.
geometry
geometry solved