The three-chord lemma relates the lengths of three chords of a circle that share an endpoint. If the middle chord lies on the angle bisector of the angle between the outer two, then
 |
(1)
|
where 
and 
are the lengths of the outer chords,
is the length
of the middle chord, and 
is either of the two equal angles
shown above (Price 1994).
More generally, let four distinct points 
, 
, 
, and 
occur in that order around a circle.
Write 
,
,
,
,
and 
,
where 
,
,
and 
.
The middle chord has length
 |
(2)
|
Both formulas follow from Ptolemy's theorem applied to the cyclic quadrilateral 
, which gives
 |
(3)
|
If 
is the radius of the circle,
then the inscribed angles give 
, 
, and 
. Substitution yields the general formula. Taking 
and using 
recovers the three-chord lemma.
