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intersecting secants theorem — Definition, Formula & Examples

The intersecting secants theorem states that when two secant lines are drawn from the same external point to a circle, the product of one secant's external segment and its whole length equals the product of the other secant's external segment and its whole length.

If two secants from an external point PP intersect a circle at points AA, BB and CC, DD respectively, then PA⋅PB=PC⋅PDPA \cdot PB = PC \cdot PD, where PAPA and PCPC are the distances from PP to the nearer intersection points and PBPB and PDPD are the distances from PP to the farther intersection points.

Key Formula

PA⋅PB=PC⋅PDPA \cdot PB = PC \cdot PD
Where:
  • PP = The external point where both secants originate
  • PAPA = Distance from P to the nearer intersection point on the first secant
  • PBPB = Distance from P to the farther intersection point on the first secant
  • PCPC = Distance from P to the nearer intersection point on the second secant
  • PDPD = Distance from P to the farther intersection point on the second secant

How It Works

Draw two secant lines from a single point outside the circle. Each secant crosses the circle at two points, creating an external segment (from the external point to the near side of the circle) and a whole segment (from the external point all the way through to the far side). Multiply the external segment by the whole segment for each secant — the two products are always equal. This relationship holds because the two secants form similar triangles through the circle's inscribed angle properties.

Worked Example

Problem: Two secants are drawn from an external point P. The first secant has an external segment of 4 and passes through the circle with a chord length of 6 (so the whole secant length is 10). The second secant has an external segment of 5. Find the whole length of the second secant.
Step 1: Identify the segments. For the first secant: external = 4, whole = 4 + 6 = 10. For the second secant: external = 5, whole = unknown.
PA=4,  PB=10,  PC=5,  PD=?PA = 4,\; PB = 10,\; PC = 5,\; PD = ?
Step 2: Apply the intersecting secants theorem.
4⋅10=5⋅PD4 \cdot 10 = 5 \cdot PD
Step 3: Solve for PD.
40=5⋅PD  ⟹  PD=840 = 5 \cdot PD \implies PD = 8
Answer: The whole length of the second secant is 8, meaning the chord portion inside the circle is 8 − 5 = 3.

Why It Matters

This theorem appears frequently in high school geometry proofs and standardized tests involving circles. It also generalizes to the power of a point, a unifying idea that connects secant-secant, secant-tangent, and chord-chord relationships into one framework.

Common Mistakes

Mistake: Using only the chord length inside the circle instead of the full secant length (external segment + chord).
Correction: Always multiply the external segment by the entire distance from the external point through the circle, not just the chord portion.

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