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 ${\displaystyle c_{P}}$ 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.^[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 the problem was given in an earlier examination paper.^[4]

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 ${\displaystyle |FF'|=|JJ'|}$.^[3]

There are some proofs for eyeball theorem, one of them shows that this theorem is a consequence of the Japanese theorem for cyclic quadrilaterals.^[5]

• Japanese theorem for cyclic quadrilaterals

• 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

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

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