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Kvant 2022
M2725
M2725
Part of
Kvant 2022
Problems
(1)
Two 2n-gons and perpendicular bisectors
Source: Kvant Magazine No. 11-12 2022 M2725
3/7/2023
Two equally oriented regular
2
n
2n
2
n
-gons
A
1
A
2
…
A
2
n
A_1A_2\ldots A_{2n}
A
1
A
2
…
A
2
n
and
B
1
B
2
…
B
2
n
B_1B_2\ldots B_{2n}
B
1
B
2
…
B
2
n
are given. The perpendicular bisectors
ℓ
i
\ell_i
ℓ
i
of the segments
A
i
B
i
A_iB_i
A
i
B
i
are drawn. Let the lines
ℓ
i
\ell_i
ℓ
i
and
ℓ
i
+
1
\ell_{i+1}
ℓ
i
+
1
intersect at the point
K
i
K_i
K
i
(hereafter we reduce indices modulo
2
n
2n
2
n
). Denote by
m
i
m_i
m
i
the line
K
i
K
i
+
n
K_iK_{i+n}
K
i
K
i
+
n
. Prove that
n
n{}
n
lines
m
i
m_i
m
i
intersect at one point and at that the angles between the lines
m
i
m_i
m
i
and
m
i
+
1
m_{i+1}
m
i
+
1
are equal.Proposed by Chan Quang Hung (Vietnam)
Kvant
geometry