The Intersecting Chords Theorem
Simple Explanation
When two chords of a circle intersect inside the circle at a point P, the two chords are each split into two segments — and the product of one chord's two segments always equals the product of the other chord's two segments: PA × PC = PB × PD.
Why Do We Need It?
This theorem lets you find an unknown chord segment length whenever two chords cross, without needing any angle information at all.
See It
A circle with two chords, AC and BD, crossing at an interior point P
Formula
The Intersecting Chords Theorem
PA × PC = PB × PD, for chords AC and BD intersecting at point P inside the circle
When two chords intersect inside a circle, the products of the two segments of each chord are always equal.
- P
- — the point where the two chords intersect
- PA, PC
- — the two segments the intersection point creates on the first chord
- PB, PD
- — the two segments the intersection point creates on the second chord
When to use it: Whenever two chords cross inside a circle, and you know three of the four segment lengths and need the fourth.
Worked Example
Find a missing chord segment
Chords AC and BD intersect at P inside a circle. PA=6, PC=8, PB=4. Find PD.
Why Does This Work?
Stated simply: triangles PAB and PDC are similar. ∠APB = ∠DPC (vertical angles), and ∠PAB = ∠PDC (both are inscribed angles subtending the same arc BC) — giving AA similarity. Similar triangles have proportional sides: PA/PD = PB/PC, which rearranges by cross-multiplication into exactly PA × PC = PB × PD.
Real-Life Example
Locating a point inside a circular arena
Two straight sightlines across a circular stadium floor cross at an interior point, and the distances to some of the four endpoints are already known.
The intersecting chords theorem lets an engineer or event planner find the one remaining unknown distance using only simple multiplication.
Practice
Chords intersect at P. PA=5, PC=9, PD=15. Find PB.
HardCommon mistake
Adding the segment lengths instead of multiplying them — the theorem is specifically about the PRODUCT of each chord's two segments, not their sum.
Quick Review
- PA × PC = PB × PD, for chords AC and BD meeting at interior point P.
- Proof idea: △PAB ~ △PDC by AA (vertical angles + equal inscribed angles on the same arc).
- Only applies when the chords intersect INSIDE the circle.