Tangents to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency. Unlike a secant, which crosses through the interior of the circle and intersects it at two points, a tangent grazes the circle at precisely one location without entering its interior. This single-touch property makes tangents one of the most elegant and useful concepts in Euclidean geometry, appearing in everything from the design of gears and pulleys to the mathematics of calculus. The study of tangents dates back to ancient Greece, where mathematicians like Euclid formally proved fundamental properties about lines and circles. One of the most important of these is the tangent-radius theorem: at the point of tangency, the tangent line is always perpendicular to the radius drawn to that point. This 90-degree relationship is not just a coincidence — it can be rigorously proved using the shortest-distance argument, and it unlocks a powerful set of tools for solving geometric problems involving circles. From a single external point, exactly two tangent lines can be drawn to a circle, and the two tangent segments from that external point to their respective points of tangency are always equal in length. This equal-tangent property has direct applications in constructions, proofs, and real-world engineering. Understanding tangents requires combining knowledge of right triangles, the Pythagorean theorem, and circle properties — making them a perfect topic for practising deductive reasoning.

Key takeaways

Definition of a Tangent
Tangent-Radius Theorem (Perpendicularity)
Equal Tangent Segments from an External Point
Angle Between a Tangent and a Chord
Common Tangents

Definition of a Tangent

A tangent to a circle is a line in the plane of the circle that intersects the circle at exactly one point. This point is called the point of tangency (or point of contact). A line that intersects a circle at two points is called a secant, and a line with no intersection points misses the circle entirely. The tangent occupies the boundary case: it touches without crossing.

Tangent-Radius Theorem (Perpendicularity)

The most fundamental property: if a line is tangent to a circle at point P, then the radius OP (where O is the centre) is perpendicular to the tangent at P, meaning the angle between them is exactly 90°. Proof sketch: suppose the radius were not perpendicular. Then we could find another point on the tangent line closer to O than P, meaning the line would cross inside the circle and intersect it at two points — contradicting it being a tangent. Therefore the radius must be perpendicular. This is a classic proof by contradiction.

Equal Tangent Segments from an External Point

If two tangent segments are drawn from an external point T to a circle with centre O, touching the circle at points A and B, then TA = TB. Proof: In triangles OAT and OBT, OA = OB (radii), OT = OT (common side), and angles OAT = OBT = 90° (tangent-radius theorem). By the RHS (Right angle-Hypotenuse-Side) congruence criterion, triangle OAT ≅ triangle OBT, so TA = TB. This result is extremely useful: it means the two tangent lengths from any external point are always equal.

Angle Between a Tangent and a Chord

The tangent-chord angle theorem states that the angle between a tangent to a circle and a chord drawn from the point of tangency equals the inscribed angle subtending the same arc on the opposite side. Formally, if tangent TPA meets chord PA at point P, the angle between the tangent and chord PA equals the inscribed angle PBA where B is any point on the major arc. This is sometimes called the Tangent-Chord Angle or the Alternate Segment Theorem, and it provides an elegant link between tangent geometry and inscribed angles.

Common Tangents

Two circles can share external or internal common tangents. If two circles do not overlap (externally separate), there are 4 common tangents: 2 external (that do not pass between the circles) and 2 internal (that cross between them). If one circle is internally tangent to the other, there is exactly 1 common tangent. If one circle is inside the other without touching, there are 0 common tangents. Counting common tangents is a useful quick test for the relative positions of two circles.

The Key Equation

TA = \sqrt{OT^2 - r^2}

TA — Tangent length (length units): The distance from the external point T to the point of tangency A along the tangent line.
OT — Distance from centre to external point (length units): The straight-line distance from the circle's centre O to the external point T. This is the hypotenuse of the right triangle OAT.
r — Radius (length units): The radius of the circle, equal to OA. Since OA ⊥ TA, triangle OAT is right-angled at A, allowing use of the Pythagorean theorem.

Worked Example

Problem. A circle has centre O and radius 5 cm. An external point T is located 13 cm from O. Find: (a) the length of the tangent TA from T to the circle, and (b) the angle OTA.

TA = 12 \text{ cm}, \quad \angle OTA = \arctan\!\left(\tfrac{5}{12}\right) \approx 22.6°

Answer. Tangent length TA = 12 cm; Angle OTA ≈ 22.6°

Figure

Diagram

Origins · 1 minEuclid of Alexandria and how this was discovered

Euclid of Alexandria

-300–-250 · Greek

Euclid's monumental work Elements (circa 300 BCE) contains the earliest rigorous treatment of tangent lines to circles. In Book III, Propositions 16–19, Euclid formally defines a tangent line and proves that the radius to the point of tangency is perpendicular to the tangent — the same theorem students use today. His axiomatic, deductive method of proof set the standard for all mathematics that followed. When King Ptolemy I asked Euclid if there was a shortcut to learning geometry, Euclid reportedly replied: 'There is no royal road to geometry.' The quote captures the spirit of rigorous mathematical thinking that has guided students ever since.

Chords and Their Properties — Chords are line segments connecting two points on a circle. The tangent-chord angle theorem directly links tangent geometry to the properties of chords, making these two concepts closely intertwined.
Inscribed Angles — The Alternate Segment Theorem states that the tangent-chord angle equals the inscribed angle in the alternate segment, creating a direct connection between tangent properties and inscribed angle theory.
The Pythagorean Theorem — Calculating the length of a tangent segment from an external point requires applying the Pythagorean theorem to the right triangle formed by the radius, the tangent, and the line from the centre to the external point.
Triangle Congruence — The proof that two tangent segments from an external point are equal relies on the RHS (Right angle-Hypotenuse-Side) congruence criterion for triangles, a key deductive reasoning tool.