Summary and Quick Revision

Chapter at a Glance

The whole of Circles rests on three facts. Fix them in memory and you can rebuild everything else.

1. Tangent ⊥ radius at the point of contact (Theorem 10.1).
2. Tangents from an external point are equal (Theorem 10.2).
3. Length of a tangent $= \sqrt{d^2 - r^2}$ ($d$ = distance of the external point from the centre, $r$ = radius).

Definitions and Positions of a Line

• Secant: a line meeting a circle at 2 points.
• Tangent: a line meeting a circle at exactly 1 point (the point of contact); it is the limiting secant.
• Non-intersecting line: 0 common points.

Number of tangents from a point: inside → 0, on → 1, outside → 2.

Key Formulae and Results (Must Memorise)

• Length of tangent: $PT = \sqrt{d^2 - r^2}$, where $d=OP$ (external point to centre).
• Right triangle: radius and tangent are the legs; centre-to-point distance is the hypotenuse, so $d^2 = r^2 + (\text{tangent})^2$.
• Chord of concentric circles (radii $R>r$) touching the inner circle: length $= 2\sqrt{R^2 - r^2}$.
• Quadrilateral circumscribing a circle: $AB + CD = AD + BC$.
• Parallelogram circumscribing a circle is a rhombus.
• Angle relation: angle between two tangents $+$ angle at the centre $= 180^\circ$; and $\angle PTQ = 2\,\angle OPQ$.
• $OP$ bisects both the angle between the tangents and the central angle.
• Tangents at the ends of a diameter are parallel.
• Triangle incircle: tangents from each vertex are equal; Area $= r \times s$ (inradius $\times$ semi-perimeter).

Common Traps to Avoid

• Do not make the tangent the hypotenuse — the distance to the centre is the hypotenuse. So the tangent length is always less than $d$.
• Angle between tangents and central angle are supplementary (sum $180^\circ$), not equal.
• A tangent cannot be drawn from a point inside the circle (0 tangents), and from a point on the circle only one.
• In "chord touches the smaller concentric circle," remember to double $\sqrt{R^2-r^2}$ for the full chord.
• For a circumscribing quadrilateral it is opposite sides whose sums are equal ($AB+CD=AD+BC$), not adjacent sides.

Final Tip: Almost every proof starts by drawing radii to the points of contact and writing "$90^\circ$" at each — do that first and the rest follows.