Eyeball theorem
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 [math]\displaystyle{ c_P }[/math] and [math]\displaystyle{ c_Q }[/math]centered at [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math] the tangents from P onto [math]\displaystyle{ c_Q }[/math] intersect [math]\displaystyle{ c_Q }[/math] at [math]\displaystyle{ C }[/math] and [math]\displaystyle{ D }[/math] and the tangents from Q onto [math]\displaystyle{ c_P }[/math] intersect [math]\displaystyle{ c_P }[/math] at [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math]. Then [math]\displaystyle{ |AB| = |CD| }[/math].
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.[3] Furthermore Evans stated that problem was given in an earlier examination paper.[4]
A variant of this theorem states, that if one draws line [math]\displaystyle{ FJ }[/math] in such a way that it intersects [math]\displaystyle{ c_P }[/math] for the second time at [math]\displaystyle{ F' }[/math] and [math]\displaystyle{ c_Q }[/math] at [math]\displaystyle{ J' }[/math]. Then, it turns out that [math]\displaystyle{ |FF'|=|JJ'| }[/math].[3]
References
- ↑ 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
- ↑ David Acheson: The Wonder Book of Geometry. Oxford University Press, 2020, ISBN 9780198846383, pp. 141–142
- ↑ 3.0 3.1 José García, Emmanuel Antonio (2022), "A Variant of the Eyeball Theorem", The College Mathematics Journal 53 (2): 147-148.
- ↑ Evans, G. W. (1938). Ratio as multiplier. Math. Teach. 31, 114–116. DOI: https://doi.org/10.5951/MT.31.3.0114.
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
- The Eyeball Theorem at cut-the-knot.org (contains a variety of proofs)
- Weisstein, Eric W.. "Eyeball Theorem". http://mathworld.wolfram.com/EyeballTheorem.html.
- Eyeball Theorem at Geometry from the Land of the Incas
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