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The central angle theorem states that the central angle subtended by two points on a circle is twice the inscribed angle subtended by those points on any part of the circle. In an equation, if the central angle is denoted by $\theta$, then the inscribed angle will be $\frac{\theta}{2}$. This theorem is fundamental in understanding the angles within a circle and how they relate to the arc they intercept.

The cyclic quadrilateral opposite angles theorem states that the sum of the opposite angles of a cyclic quadrilateral (one whose vertices all lie on a single circle) is 180 degrees ($\pi$ radians). If $A$, $B$, $C$, and $D$ are the vertices of a cyclic quadrilateral, then the theorem can be expressed as $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.

The tangent circles angles theorem states that if two tangent lines are drawn to a circle from an external point, the angle between the tangent lines is bisected by the line segment joining the external point to the center of the circle. This also implies that this bisecting line is perpendicular to the line segment connecting the points of tangency on the circle.

The intersecting chords theorem states that if two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. If the chords $AB$ and $CD$ intersect at point $X$ inside the circle, and $AX = a$, $XB = b$, $CX = c$, and $XD = d$, then the theorem tells us that $a \cdot b = c \cdot d$.

The tangent-secant theorem states that if a tangent and a secant (or another tangent) are drawn from an external point to a circle, the square of the length of the tangent segment is equal to the product of the lengths of the secant's external segment and the entire secant (external plus internal segments). Mathematically, if $t$ is the length of the tangent and $a$ and $b$ are the segments of the secant, then $t^2 = a(a + b)$.

The inscribed angle theorem states that an angle inscribed in a circle is half the measure of its intercepted arc. Therefore, if the arc measures $\alpha$ degrees, the inscribed angle will measure $\frac{\alpha}{2}$ degrees. This is true no matter where the angle is positioned on the circle.