Eyeball theorem explained

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

More precisely it states the following:^[1]

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.^[2] 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 problem was given in an earlier examination paper.^[3]

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' and c _{Q} at J' , then it turns out that |FF'| = |JJ'| .

There are some proofs for Eyeball theorem, one of them show that this theorem is a consequence of the Japanese theorem for cyclic quadrilaterals.^[4]

See also

• Japanese theorem for cyclic quadrilaterals

References

1. 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
2. David Acheson: The Wonder Book of Geometry. Oxford University Press, 2020, ISBN 9780198846383, pp. 141–142
3. Evans, G. W. (1938). Ratio as multiplier. Math. Teach. 31, 114–116. DOI: https://doi.org/10.5951/MT.31.3.0114.
4. https://www.cut-the-knot.org/Curriculum/Geometry/Eyeball.shtml The Eyeball Theorem

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