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# Secant of a Circle — Definition, Formula, and Examples

## TL;DR

A secant of a circle is a straight line that intersects the circle at two distinct points — it is a chord extended past both ends. This article covers the secant definition, how it differs from a chord and a tangent, the two power-of-a-point theorems (PA⋅PB=PC⋅PD) and the tangent–secant relation, and worked examples of a secant of a circle.

## What Is A Secant Of A Circle?

A secant of a circle is a straight line that intersects the circle at exactly two distinct points. Because it is a full line, not a segment, it extends infinitely in both directions past the circle.

The key contrast is with a **chord**, which is the segment whose two endpoints lie _on_ the circle. Take a chord and extend it both ways and you have a secant — the secant contains the chord. A **tangent** is the limiting case: slide the two intersection points of a secant together until they merge into one, and the secant becomes a tangent touching the circle at a single point.

## How Is A Secant Different From A Chord And A Tangent?

This is the question students ask most, because the three terms describe lines that look almost the same on a quick sketch. The difference is purely about how many times the line meets the circle and whether it stops.

| Line   | Meets the circle at | Extends beyond the circle? |
| ------ | ------------------- | -------------------------- |
| Chord  | 2 points            | No — it stops at the circle |
| Secant | 2 points            | Yes — it continues past both points |
| Tangent| 1 point            | Yes — it touches and continues |

A chord and a secant share the same two intersection points; the chord is just the bounded piece in the middle. A tangent meets the circle only once and sits perpendicular to the [radius](/content/math/geometry/radius/index.html) at that point.

## The secant theorems and their formulas

Secants matter because of what happens at the point where two of them meet outside the circle. These relationships are forms of the **power of a point** — a result first studied through the work of [Jakob Steiner](https://mathshistory.st-andrews.ac.uk/Biographies/Steiner/) in the 1820s.

**Two-secant (intersecting secants) theorem.** From an external point PPP, draw two secants. The first crosses the circle at AAA (near) and BBB (far); the second at CCC (near) and DDD (far). Then:

PA⋅PB=PC⋅PD

Each product multiplies the whole secant length by its external part, and the two products are equal.

**Tangent–secant theorem.** From an external point PPP, draw one tangent touching at TTT and one secant crossing at AAA (near) and BBB (far). Then:

PT²=PA⋅PB

The tangent length squared equals the secant's external part times its whole length. This is the two-secant theorem with the tangent treated as a secant whose two points have merged.

| Symbol  | Meaning |
| ------- | ------- |
| PPP    | The external point where the lines meet |
| PA, PC | Distance from PPP to the _near_ intersection |
| PB, PD | Distance from PPP to the _far_ intersection |
| PT    | Length of the tangent segment from PPP to the point of contact |

## Examples of Secant of a Circle

### Example 1

**A line meets a circle at points MMM and NNN and continues past both. Is it a chord, a secant, or a tangent?**

The line meets the circle at two points and extends beyond them. A chord would stop at the circle; a tangent would touch only once.

Final answer: it is a secant.

### Example 2

**From an external point PPP, two secants give PA=4, PB=9, and PC=3; a student finds PD=9−3=6, so find the correct PD.**

The first instinct is to subtract, treating the segments as if they simply add and remove along one line. Test it: with PD=6, the second product is PC⋅PD=3×6=18, while the first is PA⋅PB=4×9=36, so the two products are not equal and the subtraction approach is wrong.

The correct method uses the two-secant theorem:

PA⋅PB=PC⋅PD → 4×9=3×PD → PD=12.

Final answer: PD=12.

### Example 3

**Two secants from an external point give PA=5, PB=12, and PC=6. Find PD.**

Apply the two-secant theorem:

PA⋅PB=PC⋅PD → 5×12=6×PD → PD=10.

Final answer: PD=10.

### Example 4

**From a point PPP outside a circle, a tangent of length PT=8 touches the circle, and a secant from PPP has near point A with PA=4. Find the whole secant length PB.**

Use the tangent–secant theorem:

PT²=PA⋅PB → 8²=4×PB → PB=16.

Final answer: PB=16.

### Example 5

**A secant from external point PPP crosses a circle at A and B with PA=3 and AB=5. Find the length of the tangent PT from the same point.**

First, find the whole secant length PB:

PB=PA+AB=3+5=8.

Then apply the tangent–secant theorem:

PT²=PA⋅PB → PT²=3×8 → PT=√24 ≈ 4.9.

Final answer: PT≈4.9.

### Example 6

**An angle formed by two secants meeting outside a circle equals half the difference of the two intercepted arcs. If the far arc is 110° and the near arc is 40°, find the angle at the external point.**

The external-angle rule for two secants is:

∠P=0.5(far arc−near arc) → ∠P=0.5(110°−40°) → ∠P=35°.

Final answer: ∠P=35°.

## Why Secants Are Worth Defining Separately

A natural question: if a secant is just an extended chord, why give it its own name? Because the _external_ point is where the useful mathematics lives. A chord tells you about the inside of the circle; a secant lets you reason about a point sitting outside it, using only lengths you can measure from that point.

## Slip-Free Secant Work: The Mistakes To Watch

### Mistake 1: Multiplying the wrong segment lengths

**Where it slips in:** Applying the two-secant theorem when the problem gives the _outside_ piece and the _chord_ piece separately, not the full length.

**Don't do this:** Write PA⋅AB=PC⋅CD using the interior chord pieces.

**The correct way:** Each side of PA⋅PB=PC⋅PD is (external segment) × (whole secant).

### Mistake 2: Subtracting instead of using the product relationship

**Where it slips in:** When the unknown looks like it could be found by simple addition or subtraction along the line.

**Don't do this:** Treat PA, PB, PC, PD as if they combine linearly.

**The correct way:** The relationship is multiplicative — equal _products_, not equal sums.

### Mistake 3: Confusing the tangent–secant form

**Where it slips in:** Mixing the tangent length with the full secant on the wrong side.

**Don't do this:** Write PT²=AB² or PT=PA⋅PB without the square.

**The correct way:** It is PT²=PA⋅PB — the tangent length is _squared_.

## Conclusion

- A **secant of a circle** is a line that cuts the circle at two distinct points and extends beyond both — a chord extended.

- A chord stops at the circle, a secant continues past it, and a tangent meets the circle once.

- The two-secant theorem gives PA⋅PB=PC⋅PD from an external point.

- The tangent–secant theorem gives PT²=PA⋅PB.

- These are product relationships, not differences — and they describe the circle from a point outside it.

## Frequently Asked Questions

**Can a secant of a circle pass through the center?**

Yes. A secant of a circle through the center contains the diameter — the diameter is the chord on that secant, and it is the longest possible chord.

**Is every chord part of a secant?**

Yes. Extend any chord beyond both endpoints and you get the secant that contains it.

**How many times can a straight line cross a circle?**

At most twice. A line can miss the circle entirely (no intersection), touch it once (a tangent), or cross it twice (a secant). It can never meet a circle at three points.

**What is the secant line in calculus — is it the same thing?**

The names are related. In calculus, a secant line joins two points on a curve, and as those points slide together it becomes the tangent line — the same merging idea you see when a circle's secant collapses into a tangent.