Tangents and Intersecting Chords

337. The Chord Intersection Secret · PA × PB always equals PC × PD

The product of the chord segments is always equal, no matter where P sits inside.

When two chords intersect at a point P inside a circle, the products of their segments are equal: PA × PB = PC × PD. This product is the same for every chord drawn through P.

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Selina ICSE: Tangents and Intersecting Chords

What this lesson covers

Try to break it

Drag P around inside the circle. PA, PB, PC, PD change individually, but PA × PB always equals PC × PD. Push P close to the boundary or to the centre — the equality holds. Try to find a P where the products disagree; impossible. (At the centre, all four pieces are equal to r, so both products equal r².)

How you build it

Draw two intersecting chords through P.

Draw a circle and mark a point P inside it.
Draw a chord AB passing through P.
Draw another chord CD passing through P.

The proof, step by step

Prove that the products of the segments of two intersecting chords are equal.

In Δ APC and Δ BPD, ∠A = ∠D (Angles in the same segment)
∠C = ∠B (Angles in the same segment)
⇒ Δ APC ~ Δ BPD (By A.A. Postulate)
⇒ PA/PD = PC/PB (Corresponding sides of similar triangles)
⇒ PA × PB = PC × PD (Cross-multiplication)

Worked example

Two chords AB and CD of a circle intersect at a point P inside the circle. If PA = 4 cm, PB = 6 cm, and PC = 3 cm, find the length of PD.

By the Intersecting Chords Theorem, PA × PB = PC × PD. Substituting the given values: 4 × 6 = 3 × PD. Therefore, PD = 24 / 3 = 8 cm.

4 cm
6 cm
8 cm — correct
12 cm