When two chords intersect inside a circle, their segment products are always equal.
The Mnemonic
Two Crossing Chords, One Equal Product
When two chords (segments connecting two points on a circle) intersect INSIDE the circle at some point P, each chord gets split into two pieces. The Chord Product Theorem states: the product of one chord's two pieces always equals the product of the other chord's two pieces.
If chord one is split into pieces a and b, and chord two is split into pieces c and d, then: a · b = c · d — always, regardless of where inside the circle the two chords happen to cross.
Two chords crossing inside the circle at P — the two pieces of each chord multiply to the same product
💡 Memory Trick
"When two chords intersect inside a circle, their segment products are equal." Split each chord at the crossing point into two pieces, multiply each chord's own two pieces together, and those two products will always match — a·b = c·d, no matter the angle or position of the crossing.
Why It Works
A Proof Using Similar Triangles
Connect the four endpoints of the two chords with two new segments (forming an X-shaped pair of triangles sharing the intersection point). The two resulting triangles turn out to be similar by AA: they share a pair of vertical angles at the intersection point (always equal), and they share a pair of inscribed angles that intercept the SAME arc (equal by the Inscribed Angle Theorem).
Since the triangles are similar, their corresponding sides are proportional — setting up that proportion and cross-multiplying produces exactly a·b = c·d. This is a genuinely elegant connection: the Chord Product Theorem is really Triangle Similarity and the Inscribed Angle Theorem working together, not an independent rule.
Using It In A Proof
Setting Up the Equation Correctly
The main skill is correctly identifying which four segments belong to which chord, since the theorem only works when pairing each chord's OWN two pieces together — not mixing pieces from different chords.
1
Identify the two chords and their intersection point
Confirm both chords actually cross INSIDE the circle (not outside, which uses a different theorem — Power of a Point, covered next).
2
Label each chord's two segments
Each chord is split into two pieces by the intersection point — label these clearly to avoid mixing up which pieces belong to which chord.
3
Set up the equation and solve
Multiply each chord's own two pieces, set the two products equal, and solve for any unknown.
Full Worked Example
Solving for an Unknown Chord Segment
Given: Two chords intersect inside a circle. One chord is split into pieces of 6 and 8. The other chord is split into pieces of 4 and x. Find: x.
1
Set up the equation
By the Chord Product Theorem: 6 × 8 = 4 × x.
2
Simplify the left side
48 = 4x.
3
Solve for x
x = 12.
Always double-check that you've correctly identified which two numbers belong to the SAME chord before setting up this equation — a common error is accidentally pairing pieces from different chords.
🎯 Quick Worked Example
Two chords intersect inside a circle. One is split into pieces of 5 and 9. The other is split into two equal pieces of length y. Find y.
1
Set up the equation. 5 × 9 = y × y.
2
Simplify. 45 = y².
3
Solve. y = √45 = 3√5 ≈ 6.7.
📌 Exam Application
Watch for problems where one chord's pieces are given as algebraic expressions rather than plain numbers — these require setting up and solving a genuine equation (sometimes quadratic, if the unknown appears on both sides) rather than simple arithmetic division.
⚠️ Most Common Chord Product Theorem Mistakes
Trap 1 — Mixing up which pieces belong to which chord: Only multiply the two pieces that belong to the SAME chord together — pairing a piece from chord one with a piece from chord two produces a meaningless equation.
Trap 2 — Using this theorem for chords that don't intersect INSIDE the circle: If the two chords (or their extensions) meet OUTSIDE the circle instead, a different theorem (Power of a Point) applies — this theorem specifically requires the intersection point to be inside the circle.
✓ Quick Self-Test
1) State the Chord Product Theorem. 2) Two chords intersect with one split into 3 and 10, the other split into 5 and x — find x. 3) What proof technique (from an earlier lesson) is used to prove this theorem? 4) Why must the two chords intersect INSIDE the circle for this theorem to apply? 5) Two chords intersect, with one split into equal pieces of 6 each — if the other chord is split into 4 and z, find z.