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The eyeball theorem is a statement in elementary geometry about a property of a pair of disjoint 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 and centered at and , label the points where the tangent lines from P onto cross as and , and label the points where the tangents from Q onto cross as and . Then .
The eyeball theorem was discovered in the 1960s 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 George W. Evans in 1938. Evans stated that the problem was given in "some examination paper".
A related theorem states that if and are points of tangency from on and from on respectively, and one draws line in such a way that it intersects for the second time at and at , then .
Several proofs are known; one derives the theorem from the Japanese theorem for cyclic quadrilaterals.The Eyeball Theorem at cut-the-knot.org
See also Japanese theorem for cyclic quadrilaterals
References
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