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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 ${\displaystyle c_{P}}$ and ${\displaystyle c_{Q}}$centered at ${\displaystyle P}$ and ${\displaystyle Q}$ the tangents from P onto ${\displaystyle c_{Q}}$ intersect ${\displaystyle c_{Q}}$ at ${\displaystyle C}$ and ${\displaystyle D}$ and the tangents from Q onto ${\displaystyle c_{P}}$ intersect ${\displaystyle c_{P}}$ at ${\displaystyle A}$ and ${\displaystyle B}$. Then ${\displaystyle |AB|=|CD|}$.

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 problem was given in an earlier examination paper.

A variant of this theorem states that if one draws line ${\displaystyle FJ}$ in such a way that it intersects ${\displaystyle c_{P}}$ for the second time at ${\displaystyle F'}$ and ${\displaystyle c_{Q}}$ at ${\displaystyle J'}$, then it turns out that ${\displaystyle |FF'|=|JJ'|}$.

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

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. ^ José García, Emmanuel Antonio (2022), "A Variant of the Eyeball Theorem", The College Mathematics Journal, 53 (2): 147-148.
4. Evans, G. W. (1938). Ratio as multiplier. Math. Teach. 31, 114–116. DOI: https://doi.org/10.5951/MT.31.3.0114.
5. The Eyeball Theorem at cut-the-knot.org

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

• Weisstein, Eric W. "Eyeball Theorem". MathWorld.
• Eyeball Theorem at Geometry from the Land of the Incas

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