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4
4
Part of
2021 Stanford Mathematics Tournament
Problems
(1)
SMT 2021 Geometry #4
Source:
8/9/2023
△
A
0
B
0
C
0
\vartriangle A_0B_0C_0
△
A
0
B
0
C
0
has side lengths
A
0
B
0
=
13
A_0B_0 = 13
A
0
B
0
=
13
,
B
0
C
0
=
14
B_0C_0 = 14
B
0
C
0
=
14
, and
C
0
A
0
=
15
C_0A_0 = 15
C
0
A
0
=
15
.
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
is inscribed in the incircle of
△
A
0
B
0
C
0
\vartriangle A_0B_0C_0
△
A
0
B
0
C
0
such that it is similar to the first triangle. Beginning with
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
, the same steps are repeated to construct
△
A
2
B
2
C
2
\vartriangle A_2B_2C_2
△
A
2
B
2
C
2
, and so on infinitely many times. What is the value of
∑
i
=
0
∞
A
i
B
i
\sum_{i=0}^{\infty} A_iB_i
∑
i
=
0
∞
A
i
B
i
?
geometry