The praying eyes theorem is an equality between two chords constructed from tangents drawn from the centers
of two circles. Let the circles
have centers 
,
,
radii 
, 
, and center distance 
. Choose 
on the first circle so that 
is a tangent
line, and 
on the second so that 
is a tangent line. If the
line 
meets the first and second circles
again at 
and 
,
respectively, then
 |
(1)
|
The equality holds for either choice of each tangent line (E. A. José García, pers. comm., Aug. 15, 2014).
For a proof, assign oriented coordinates 
, 
, 
, and 
to 
, 
, 
, and 
on 
. The Pythagorean theorem
and circle power give
 |  |  |
(2)
|
 |  |  |
(3)
|
Thus 
,
which implies 
(Bogomolny 2013).
Replacing the two circles by spheres with the same centers and radii gives a three-dimensional
extension (José García 2014). The plane through
perpendicular to the original plane
cuts the spheres in circles
whose centers lie on 
. Their diameters are the equal
chords 
and 
, so the sections are congruent.
