Geometry - Circles (Angle properties, Cyclic properties, Tangent and Secant properties)

The angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle. Visually, if you have an arc $AB$ and a center $O$, drawing lines $OA$ and $OB$ creates a central angle $\angle AOB$. If you pick any point $P$ on the circle's circumference and connect $PA$ and $PB$, the resulting angle $\angle APB$ will satisfy the relationship $\angle AOB = 2\angle APB$.

Angles in the same segment of a circle are equal. This can be visualized by drawing a chord $AB$ and picking two points $C$ and $D$ on the same side of the chord on the circumference. The angles $\angle ACB$ and $\angle ADB$ will be identical because they are subtended by the same arc $AB$.

The angle in a semi-circle is a right angle ($90^\circ$). If you draw a diameter $AB$ passing through the center $O$ and pick any point $P$ on the circumference, the triangle formed by $A, B,$ and $P$ will always have $\angle APB = 90^\circ$.

In a cyclic quadrilateral, the opposite angles are supplementary, meaning they add up to $180^\circ$. Visually, if a quadrilateral $ABCD$ has all four vertices lying on the circumference of a circle, then $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$. Additionally, any exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

The tangent at any point of a circle is perpendicular to the radius through the point of contact. If a line touches the circle at point $P$ and $O$ is the center, the radius $OP$ and the tangent line meet to form a $90^\circ$ angle ($OP \perp$ tangent).

From any external point, exactly two tangents can be drawn to a circle, and these tangents are equal in length. If point $P$ is outside the circle and $PA$ and $PB$ are the tangents touching the circle at $A$ and $B$ respectively, then $PA = PB$. Also, the line $OP$ bisects the angle $\angle APB$ and $\angle AOB$.

The Alternate Segment Theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. If a tangent $XY$ touches the circle at point $A$ and a chord $AB$ is drawn, the angle $\angle BAX$ is equal to the angle subtended by chord $AB$ at any point $C$ in the major segment ($\angle BAX = \angle BCA$).

Intersecting Chords and Secant Properties: When two chords $AB$ and $CD$ intersect inside a circle at point $P$, the product of the lengths of their segments is equal ($PA \times PB = PC \times PD$). If the chords intersect externally (secants), the same product rule applies to the segments measured from the external point $P$ to the circle's boundaries.