Division of Circle into 4 Equal Parts by 3 Equal Length Lines

Problem

 * Divide a circle into $4$ equal parts by $3$ lines of equal length.

By equal here, we mean of equal area.

Solution

 * Circle-into-4-by-3-equal-lines.png

Each of the curved lines is a semicircle whose radius is $\dfrac {n r} 4$, where:
 * $n$ goes from $1$ to $3$
 * $r$ is the radius of the main circle.

Proof
Let the main circle have radius $4 a$, to make the numbers convenient.

Let the area of the main circle be $\AA$.

From Area of Circle:


 * $\AA = \pi \paren {4 a}^2 = 16 \pi a^2$

The area $\AA_1$ of one of the outside shapes is:


 * half the area of the main circle
 * less half the area of the circle whose radius is $3 a$
 * plus half the area of the circle whose radius is $a$

That is:

The area $\AA_2$ of one of the inside shapes is:


 * half the area of the circle whose radius is $3 a$
 * less half the area of the circle whose radius is $2 a$
 * plus half the area of the circle whose radius is $2 a$
 * less half the area of the circle whose radius is $a$

That is: