Tangents from a Point

A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency. One of the most elegant results in Euclidean geometry concerns how many tangent lines can be drawn to a circle from a point not on the circle. The answer depends entirely on where the point is located relative to the circle: if the point lies outside the circle, exactly two tangents can be drawn; if it lies on the circle, exactly one tangent exists; and if it lies inside the circle, no real tangent can be drawn at all. The two tangent segments drawn from an external point to a circle are always equal in length. This is not just an observation — it is a theorem that can be rigorously proved using congruent triangles or the Pythagorean theorem. This property makes tangent lines from an external point extremely useful in solving geometric problems, especially those involving circles inscribed in polygons or mixed perimeter calculations. Understanding tangents from an external point sits at the heart of deductive reasoning in geometry. Starting from a small set of definitions and axioms, we can chain logical steps together to arrive at powerful conclusions. The equal-tangent theorem is a perfect example of how abstract reasoning produces concrete, measurable results — a hallmark of Euclidean geometry that has been celebrated since antiquity.

Key takeaways
  • Setting Up the Configuration
  • The Right-Angle Condition
  • Counting the Tangent Lines
  • Proving the Two Tangent Segments Are Equal
  • Special Cases

Setting Up the Configuration

Let circle ω have centre O and radius r. Let P be a point external to the circle, so OP > r. A line through P is tangent to the circle if and only if it meets the circle at exactly one point. Geometrically, this happens when the distance from the centre O to the line equals exactly r.

The Right-Angle Condition

A fundamental theorem states: a tangent to a circle is perpendicular to the radius drawn to the point of tangency. So if T is the point of tangency, then OT ⊥ PT. This means triangle OTP is right-angled at T. Using the Pythagorean theorem: PT² = OP² − OT² = OP² − r². Since OP > r, this value is positive, giving a unique positive length PT.

Counting the Tangent Lines

The equation PT² = OP² − r² tells us the length of any tangent segment from P to a tangency point T. There are exactly two points T₁ and T₂ on the circle that satisfy OT ⊥ PT (one on each side of the line OP). Each gives one tangent line, so exactly two tangent lines pass through P. Rotating the right triangle OTP about OP confirms there are precisely two solutions.

Proving the Two Tangent Segments Are Equal

Consider triangles OT₁P and OT₂P. We have: OT₁ = OT₂ = r (both are radii), OP = OP (common hypotenuse), and ∠OT₁P = ∠OT₂P = 90°. By the Hypotenuse-Leg (HL) congruence theorem, △OT₁P ≅ △OT₂P. Therefore PT₁ = PT₂. The two tangent segments from an external point are equal in length.

Special Cases

When P lies on the circle, OP = r, so PT² = r² − r² = 0, meaning T = P and there is exactly one tangent — the line perpendicular to the radius at P. When P is inside the circle, OP < r, so PT² = OP² − r² < 0, which has no real solution, confirming that no tangent can be drawn from an interior point.

The Key Equation

PT=OP2−r2PT = \sqrt{OP^2 - r^2}
  • PT — Tangent segment length (length units): The distance from the external point P to the point of tangency T on the circle.
  • OP — Distance from external point to centre (length units): The distance from point P to the centre O of the circle; must satisfy OP > r for the point to be external.
  • r — Radius (length units): The radius of the circle, equal to OT, the distance from the centre O to the point of tangency T.

Worked Example

Problem. A circle has centre O and radius 5 cm. A point P is located 13 cm from O. (a) How many tangents can be drawn from P to the circle? (b) Find the length of each tangent segment. (c) If T₁ and T₂ are the two points of tangency, find the length of the chord T₁T₂.

PT1=PT2=12 cm,T1T2=12013 cmPT_1 = PT_2 = 12\text{ cm},\quad T_1T_2 = \dfrac{120}{13}\text{ cm}

Answer. Two tangents can be drawn; each tangent segment is 12 cm; chord T₁T₂ = 120/13 cm ≈ 9.23 cm.

Figure
Diagram
OPOP = 13T₁T₂r=5r=5PT₁ = PT₂ = 12T₁T₂Two equal tangent segments from external point P∠OT₁P = ∠OT₂P = 90° (radius ⊥ tangent)
Origins · 1 minEuclid of Alexandria and how this was discovered
EO
Euclid of Alexandria
-325–-265 · Greek

Euclid systematised geometry in his monumental work Elements (c. 300 BCE), a 13-book treatise that remained the definitive geometry textbook for over 2,000 years. Book III of the Elements deals specifically with circles, including propositions about tangent lines, chords, and the angles they form. Proposition III.16 establishes that the tangent to a circle is perpendicular to the diameter at the point of contact, and Proposition III.17 shows how to construct a tangent from an external point — the theoretical backbone of our topic today. 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 phrase has echoed through centuries as a reminder that rigorous thinking cannot be bypassed.

  • Tangent–Radius Perpendicularity — The proof that tangent segments from an external point are equal depends entirely on the theorem that a radius to a point of tangency is perpendicular to the tangent line.
  • Hypotenuse-Leg (HL) Congruence Theorem — The HL congruence theorem is the key deductive step used to prove that PT₁ = PT₂, making it essential background knowledge for this topic.
  • Chord Properties of a Circle — The chord T₁T₂ connecting the two points of tangency appears naturally in tangent problems; its length and bisection by OP link tangent theory to chord geometry.
  • Circles Inscribed in Polygons — The equal-tangent theorem is used repeatedly when a circle is inscribed in a triangle or polygon, allowing perimeter calculations from tangent segment lengths.