Eyeball theorem

File:Eyeball theorem variation.svg

Eyeball theorem: red chords are of equal length.
Variant: blue chords are of equal length.

The eyeball theorem is a statement in elementary geometry about a property of a pair of disjoined circles.

More precisely it states the following:

For two nonintersecting circles $c_P$ and $c_Q$centered at $P$ and $Q$ the tangents from P onto $c_Q$ intersect $c_Q$ at $C$ and $D$ and the tangents from Q onto $c_P$ intersect $c_P$ at $A$ and $B$. Then $|AB|=|CD|$.

The eyeball theorem was discovered in 1960 by the Peruvian mathematician Antonio Gutierrez. However, without the use of its current name it was already posed and solved as a problem in an article by G. W. Evans in 1938. Furthermore, Evans stated that the problem was given in an earlier examination paper.

A variant of this theorem states that if one draws line $FJ$ in such a way that it intersects $c_P$ for the second time at $F^{\prime }$ and $c_Q$ at $J^{\prime }$, then it turns out that $|F{F}^{\prime }|=|J{J}^{\prime }|$.

Several proofs are known; one derives the theorem from the Japanese theorem for cyclic quadrilaterals.

See also

• Japanese theorem for cyclic quadrilaterals

Further reading

• Antonio Gutierrez: Eyeball theorems. In: Chris Pritchard (ed.): The Changing Shape of Geometry. Celebrating a Century of Geometry and Geometry Teaching. Cambridge University Press, 2003, ISBN 9780521531627, pp. 274–280

External links

• Eyeball Theorem at Geometry from the Land of the Incas