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\begin{document}

%-------------------------------------------------------------------

\section*{intersecting chords theorem}

\hypertarget{intersecting_chords_theorem}{}\section*{{Intersecting Chords Theorem}}\label{intersecting_chords_theorem}

The \textbf{intersecting chords theorem} relates the lengths of the pieces of two non-parallel chords drawn in a circle. The chords are broken at their intersection point, which might be inside the circle or which might require the chords to be extended outside the circle. One of the chords can also be a [[tangent]] to the circle.

The intersecting chords theorem is equivalent to [[Pythagoras' Theorem]].

\hypertarget{intersection_inside}{}\subsection*{{Intersection Inside}}\label{intersection_inside}

[[IntersectingChordsInside.png:pic]]

With the chords $A B$ and $C D$ as above, and the intersection point $P$ inside the circle, then the intersecting chords theorem states that:

\begin{displaymath}
A P \times P B = C P \times P D
\end{displaymath}
\hypertarget{intersection_outside}{}\subsection*{{Intersection Outside}}\label{intersection_outside}

[[IntersectingChordsOutside.png:pic]]

With the chords $A B$ and $C D$ as above, and the intersection point $P$ outside the circle formed by extending the chords to their intersection, then the intersecting chords theorem again states that:

\begin{displaymath}
A P \times P B = C P \times P D
\end{displaymath}
\hypertarget{tangential_case}{}\subsection*{{Tangential Case}}\label{tangential_case}

[[IntersectingChordsTangent.png:pic]]

With the chord $A B$ and a tangent at a point $C$ as above, and the intersection point $P$ of the chord and tangent formed by extending the chord to this intersection, then the intersecting chords theorem states that:

\begin{displaymath}
A P \times P B = C P^2
\end{displaymath}
category: circle theorems

[[!redirects intersecting chords]]



\end{document}