The eyeball theorem states that, for two circles with each center outside the other circle, the two tangent rays from each center to the other circle
cut equal chords 
and 
on the circle with that center.
Let 
and 
be the radii and 
the distance between the centers.
If 
is half the angle between the tangent rays from the first center, then a right
triangle gives 
.
The chord length is therefore
 |
(1)
|
Exchanging the two circles gives the same expression for 
(P. Weisenhorn, pers. comm., Sep. 10,
2017).
The praying eyes theorem gives equal chords from a different choice of the secant through the two circles.
The illustrated configuration can also be described by the nine equations
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(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
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(6)
|
 |  |  |
(7)
|
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(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
Eliminating 
,
, 
, 
, 
, 
, and 
with a Gröbner basis
gives identical eighth-degree polynomial equations
for 
and 
.
