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Problem 5.16 (Star Theorem)
Problem 5.16 (Star Theorem)
Source:
May 27, 2018
Chapter 5
Problem Statement
Let
A
1
A
2
A
3
A
4
A
5
A_1A_2A_3A_4A_5
A
1
A
2
A
3
A
4
A
5
be a convex pentagon. Suppose rays
A
2
A
3
A_2A_3
A
2
A
3
and
A
5
A
4
A_5A_4
A
5
A
4
meet at the point
X
1
X_1
X
1
. Define
X
2
X_2
X
2
,
X
3
X_3
X
3
,
X
4
X_4
X
4
,
X
5
X_5
X
5
similarly. Prove that
∏
i
=
1
5
X
i
A
i
+
2
=
∏
i
=
1
5
X
i
A
i
+
3
\displaystyle\prod_{i=1}^{5} X_iA_{i+2} = \displaystyle\prod_{i=1}^{5} X_iA_{i+3}
i
=
1
∏
5
X
i
A
i
+
2
=
i
=
1
∏
5
X
i
A
i
+
3
where the indices are taken modulo 5.
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