Intersecting Chords Theorem

Statement

The Intersecting Chords Theorem states that if two chords intersect inside a circle, then the products of the lengths of the two segments of one chord and the two segments of the other chord are equal:

\[ (AP)(PB) = (CP)(PD) \]

Here, two chords \(AB\) and \(CD\) intersect at point \(P\) inside the circle.

Why it’s true

• The triangles \( \triangle APD \) and \( \triangle CPB \) formed at the intersection are similar (angle–angle reasoning, using opposite angles and angles in the same segment).
• Similarity of triangles implies a ratio of sides, which leads to the product relationship between the chord segments.
• This equality always holds, regardless of where the intersection point is inside the circle.

Recipe (how to use it)

1. Identify the intersecting chords inside the circle.
2. Label the four segments as \(AP\), \(PB\), \(CP\), and \(PD\).
3. Use the relation \(AP \times PB = CP \times PD\).
4. Solve for the unknown length if three of the segments are known.

Spotting it

You use this theorem whenever two chords cross inside a circle and you are given three of the segment lengths, with the fourth to be found.

Common pairings

• Circle theorems involving tangents and secants.
• Applications with power of a point theorem.
• Problems requiring algebraic solutions in coordinate geometry.

Mini examples

1. Given: \(AP=2\), \(PB=6\), \(CP=3\). Find: \(PD\). Answer: \(2 \times 6 = 3 \times PD \Rightarrow PD=4\).
2. Given: \(AP=5\), \(PB=4\), \(PD=10\). Find: \(CP\). Answer: \(5 \times 4 = CP \times 10 \Rightarrow CP=2\).

Pitfalls

• Confusing which segments to multiply: Always take the two parts of the same chord and multiply them.
• Forgetting it only works inside the circle: A different formula applies for tangents and secants outside the circle.
• Arithmetic slips: Check your multiplication and division carefully.

Exam strategy

• Draw a clear diagram, marking all segment lengths.
• Use the formula as soon as you see intersecting chords inside a circle.
• If algebra is involved, set up the equation carefully before solving.

Summary

The Intersecting Chords Theorem says that for two intersecting chords inside a circle, \(AP \times PB = CP \times PD\). It provides a powerful way to calculate missing lengths and is a core fact in circle geometry, often used in combination with other theorems.