This is for a real life situation not a theoretical one. I'm trying to check if a point exist in a Segment of a 2d circle. In other words, I need to know if a given point P(x,y) is anywhere inside the blue part of the circle (see image below).
Given The radius, the center $(x, y)$, point $A(x,y)$, point $B(x,y)$. The point $P(x,y)$ is also given and I need to know if it's inside or outside the Circle's segment.
What I don't know The Arc Length and the Center Angle of the Sector.