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South Africa Contests
South Africa National Olympiad
1997 South africa National Olympiad
1
1
Part of
1997 South africa National Olympiad
Problems
(1)
Sequence of triangles
Source: South Africa 1997
10/8/2005
From an initial triangle
Δ
A
0
B
0
C
0
\Delta A_0B_0C_0
Δ
A
0
B
0
C
0
, a sequence of triangles
Δ
A
1
B
1
C
1
\Delta A_1B_1C_1
Δ
A
1
B
1
C
1
,
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
, ... is formed such that, at each stage,
A
k
+
1
A_{k + 1}
A
k
+
1
,
B
k
+
1
B_{k + 1}
B
k
+
1
and
C
k
+
1
C_{k + 1}
C
k
+
1
are the points where the incircle of
Δ
A
k
B
k
C
k
\Delta A_kB_kC_k
Δ
A
k
B
k
C
k
touches the sides
B
k
C
k
B_kC_k
B
k
C
k
,
C
k
A
k
C_kA_k
C
k
A
k
and
A
k
B
k
A_kB_k
A
k
B
k
respectively. (a) Express
∠
A
k
+
1
B
k
+
1
C
k
+
1
\angle A_{k + 1}B_{k + 1}C_{k + 1}
∠
A
k
+
1
B
k
+
1
C
k
+
1
in terms of
∠
A
k
B
k
C
k
\angle A_kB_kC_k
∠
A
k
B
k
C
k
. (b) Deduce that, as
k
k
k
increases,
∠
A
k
B
k
C
k
\angle A_kB_kC_k
∠
A
k
B
k
C
k
tends to
6
0
∘
60^{\circ}
6
0
∘
.
geometry
limit