MathDB

Consider a regular heptagon ( polygon of

7

equal sides and angles)

ABCDEFG

as in the figure below:-

(a).

Prove

\frac{1}{\sin\frac{\pi}{7}}=\frac{1}{\sin\frac{2\pi}{7}}+\frac{1}{\sin\frac{3\pi}{7}}

(b).

Using

(a)

or otherwise, show that

\frac{1}{AG}=\frac{1}{AF}+\frac{1}{AE}

[asy] draw(dir(360/7)..dir(2*360/7),blue); draw(dir(2*360/7)..dir(3*360/7),blue); draw(dir(3*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(5*360/7),blue); draw(dir(5*360/7)..dir(6*360/7),blue); draw(dir(6*360/7)..dir(7*360/7),blue); draw(dir(7*360/7)..dir(360/7),blue); draw(dir(2*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(1*360/7),blue); label("

A

",dir(4*360/7),W); label("

B

",dir(5*360/7),S); label("

C

",dir(6*360/7),S); label("

D

",dir(7*360/7),E); label("

E

",dir(1*360/7),E); label("

F

",dir(2*360/7),N); label("

G

",dir(3*360/7),W); [/asy]

Problem Statement