Geometry: Circles (Chords & Tangents)

• Angle at Center: The angle subtended by an arc at the center is double the angle subtended by it at any remaining part of the circle.
• Angles in Same Segment: Angles formed by drawing lines from the endpoints of an arc to any point on the remaining arc are equal.
• Angle in a Semicircle: The angle subtended by a diameter at the circumference is always exactly 90°.
• Cyclic Quadrilateral: A 4-sided polygon inscribed in a circle. The opposite angles always sum up to 180°.

• 90° Rule: A radius or diameter is always strictly perpendicular (90°) to the tangent at the point of contact.
• External Tangents: Two tangents drawn to a circle from an external point are always equal in length ($PA = PB$). The line joining the external point to the center bisects the angle between the tangents.

When lines intersect inside or outside a circle, specific length relationships hold:

• Two Chords intersecting internally: If AB and CD intersect at P, then $PA \times PB = PC \times PD$.
• Two Secants intersecting externally: If PAB and PCD are secants cutting the circle, then $PA \times PB = PC \times PD$ (Remember: Whole Length $\times$ Outer Length).
• Tangent-Secant Theorem (The Golden Formula): If PT is tangent and PAB is a secant, then $PT^2 = PA \times PB$.

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment of the circle. This is a very common 'hidden' property in SSC CGL problems.