Intersecting Chords Theorem - Hubert Shutrick's PWW

Given a point \(P\) in the interior of a circle, pass two lines through \(P\) that intersect the circle in points \(A\) and \(D\) and, respectively, \(B\) and \(C\). Then \(AP\cdot DP = BP\cdot CP\).

Intersecting Chords Theorem - statement

If \(a=AP\), \(b=BP\), \(c=CP\), and \(d=DP\), then the Intersecting Chors Theorem is expressed as

\(a\cdot d = b\cdot c\).

(The applet below illustrates a proof by Hubert Shutrick. Points \(A\), \(B\), \(C\), \(D\), \(O\), \(R\) are draggable. Point \(O\) is the center of the given circle, \(R\) - a point on the circle.)

18 January 2015, Created with GeoGebra

Created with GeoGebra

Proof

This proof without words is due to Hubert Shutrick.

Intersecting Chords Theorem - H. Shutrick's PWW

(For an explanation - if needed, see a slightly modified version.)

This is a simple example showing how the product identities that you get algebraically from similar triangles can be illustrated. A more sophisticated example is the proof of Ptolemy's theorem with the Pythagorean theorem as the special case when the quadrilateral is a rectangle.


Power of a Point wrt a Circle

  1. Power of a Point Theorem
  2. A Neglected Pythagorean-Like Formula
  3. Collinearity with the Orthocenter
  4. Circles On Cevians
  5. Collinearity via Concyclicity
  6. Altitudes and the Power of a Point
  7. Three Points Casey's Theorem
  8. Terquem's Theorem
  9. Intersecting Chords Theorem
  10. Intersecting Chords Theorem - a Visual Proof
  11. Intersecting Chords Theorem - Hubert Shutrick's PWW

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