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Eyeball Theorem


EyeballTheorem

Given two circles, draw the tangents from the center of each circle to the sides of the other. Then the line segments AB and CD are of equal length.

The theorem can be proved by brute force by setting up the nine equations

d_1^2+h^2=h_1^2
(1)
d_2^2+h^2=h_2^2
(2)
r_1^2+(l_1-h_2)^2=h_1^2
(3)
r_2^2+(l_2-h_1)^2=h_2^2
(4)
d_1+d_2=d
(5)
r_1^2+l_1^2=d^2
(6)
r_2^2+l_2^2=d^2
(7)
s_1h_1=hr_1
(8)
s_2h_2=hr_2
(9)

and using Gröbner basis to determine the polynomial equations satisfied by s_1 and s_2 while eliminate d_1, d_2, h, h_1, h_2, l_1, and l_2. The resulting eighth-degree polynomials satisfied by s_1 and s_2 are identical, proving that AB=CD.


See also

Circle, Circle-Circle Tangents

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References

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 70, 1991.

Referenced on Wolfram|Alpha

Eyeball Theorem

Cite this as:

Weisstein, Eric W. "Eyeball Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EyeballTheorem.html

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