# 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 $AP\cdot DP = BP\cdot CP$.

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.)

Created with GeoGebra

### Proof

This proof without words is due to Hubert Shutrick.

(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.

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