Chord dividing circle , function

Question

Two chords PA and PB divide circle into three parts. The angle PAB is a root of f(x)=0.

Find f(x)

Clearly , PA and PB divides circle into three parts means it divides it into 3 parts of equal areas

How can i find f(x) then ?

thanks

Answer 1

You may assume your circle to be the unit circle in the $(x,y)$-plane and $P=(1,0)$. If the three parts have to have equal area then $A=\bigl(\cos(2\phi),\sin(2\phi)\bigr)$ and $B=\bigl(\cos(2\phi),-\sin(2\phi)\bigr)$ for some $\phi\in\ ]0,{\pi\over2}[\ $. Calculating the area over the segment $PA$ gives the condition $$2\Bigl({\phi\over2}-{1\over2}\cos\phi\sin\phi\Bigr)={\pi\over3}\ ,$$ or $f(\phi):=\phi-\cos\phi\sin\phi-{\pi\over 3}=0$. This equation has to be solved numerically. One finds $\phi\doteq1.30266$.

Answer 2

Hint: If you look up circular segment in Wikipedia, you should be able to write the area of the two segments cut off as a function of the angle between the chord and the tangent. That angle is $\theta/2$ using the figure in the article. Then the area of the circle that is left is what you want. The question is whether you can write this area as a function of the angle between the chords (one variable) instead of the angles between the chords and their respective tangents (two variables).