Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Mathematics
Mathematics
Problem
A circle is inscribed in an
n
n
n
-sided regular polygon
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \dots, A_n
A
1
,
A
2
,
…
,
A
n
having each side
a
a
a
unit. For any arbitrary point
P
P
P
on the circle, prove that
∑
i
=
1
n
(
P
A
i
)
2
=
n
a
2
4
(
1
+
cos
2
(
π
/
n
)
sin
2
(
π
/
n
)
)
.
\sum_{i=1}^{n} (PA_i)^2 = n \frac{a^2}{4} \left( \frac{1 + \cos^2(\pi/n)}{\sin^2(\pi/n)} \right).
i
=
1
∑
n
(
P
A
i
)
2
=
n
4
a
2
(
sin
2
(
π
/
n
)
1
+
cos
2
(
π
/
n
)
)
.
Browse the problem library