Inscribed angle theorem

This formula relates the inscribed angle $\angle APB$ to the central angle $\angle AOB$ on the same arc, with circle center $O$. The inscribed angle is always half the central angle; for example, if the central angle is $60^\circ$, the inscribed angle is $30^\circ$.

This rule shows that inscribed angles $\angle APB$ and $\angle AQB$ subtended by the same arc are equal. The angle remains identical wherever vertices $P$ and $Q$ lie on the circle, making it useful for comparing angles.

This rule states that when segment $AB$ is a diameter, the inscribed angle $\angle APB$ subtended by it is always a right angle ($90^\circ$). It is useful for locating right triangles inside circles.

This theorem tests whether four points $A, B, P, Q$ lie on the same circle. If points $P$ and $Q$ lie on the same side of line $AB$ and $\angle APB = \angle AQB$, then the four points are concyclic.