Given a chord of a circle, draw any other two
chords and passing through its midpoint.
Call the points where
and
meet and . Then is also the midpoint of . There are a number of proofs of this
theorem, including those by W. G. Horner, Johnson (1929, p. 78), and
Coxeter (1987, pp. 78 and 144). The latter concise proof employs projective
geometry.
The following proof is given by Coxeter and Greitzer (1967, p. 46). In the figure at right, drop perpendiculars and from and to , and and from and to . Write , , and , and then note that by similar
triangles
of a circle, draw any other two
chords 
and 
passing through its midpoint.
Call the points where 
and 
meet 
and 
. Then 
is also the midpoint of 
. There are a number of proofs of this
theorem, including those by W. G. Horner, Johnson (1929, p. 78), and
Coxeter (1987, pp. 78 and 144). The latter concise proof employs projective
geometry.
and 
from 
and 
to 
, and 
and 
from 
and 
to 
. Write 
, 
, and 
, and then note that by similar
triangles
.
Q.E.D.
