Eyeball theorem

__NOTOC__

{{Short description|Statement in elementary geometry}}

File:Eyeball theorem variation.svg

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

More precisely it states the following:Claudi Alsina, Roger B. Nelsen: Icons of Mathematics: An Exploration of Twenty Key Images. MAA, 2011, ISBN 978-0-88385-352-8, pp. 132–133

:''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. David Acheson: The Wonder Book of Geometry. Oxford University Press, 2020, ISBN 9780198846383, pp. 141–142 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.Evans, G. W. (1938). Ratio as multiplier. Math. Teach. 31, 114–116. DOI: https://doi.org/10.5951/MT.31.3.0114.

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 $F\text{'}$ and $c\_Q$ at $J\text{'}$, then it turns out that $|FF\text{'}|=|JJ\text{'}|$.{{citation

|last=José García |first=Emmanuel Antonio |title=A Variant of the Eyeball Theorem |journal=The College Mathematics Journal |volume=53 |number=2 |year=2022 |pages=147–148|doi=10.1080/07468342.2022.2022905 }}

There are some proofs for eyeball theorem, one of them shows that this theorem is a consequence of the Japanese theorem for cyclic quadrilaterals.[https://www.cut-the-knot.org/Curriculum/Geometry/Eyeball.shtml The Eyeball Theorem] at cut-the-knot.org