Haruki's Theorem

Haruki's Theorem says that given three intersecting circles that only intersect each other at two points that the lines connecting the inner intersecting points to the outer satisfy:

s_{1}\cdot s_{3}\cdot s_{5}=s_{2}\cdot s_{4}\cdot s_{6}

where $s_{1},s_{2},s_{3},s_{4},s_{5},s_{6}$ are the measure of segments connecting the inner and outer intersection points.^[1]^[2]^[3]^[4]

The theorem is named after the Japanese mathematician Hiroshi Haruki.

References

^ Haruki, Hiroshi (1972). "On a metrical theorem in geometry of circles". Colloquium Mathematicum. 25: 99–102. doi:10.4064/cm-25-1-99-102.
^ Wisstein, Eric. "Haruki's Theorem". Wolfram MathWorld. Wolfram MathWorld. Retrieved 19 August 2015.
^ Bogomolny, Alexander. "Cut the Knot". Retrieved 19 August 2015.
^ Ross Honsberger: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. MAA, 1995, p. 144-146

[1] Haruki, Hiroshi (1972). "On a metrical theorem in geometry of circles". Colloquium Mathematicum. 25: 99–102. doi:10.4064/cm-25-1-99-102.

[2] Wisstein, Eric. "Haruki's Theorem". Wolfram MathWorld. Wolfram MathWorld. Retrieved 19 August 2015.

[3] Bogomolny, Alexander. "Cut the Knot". Retrieved 19 August 2015.

[4] Ross Honsberger: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. MAA, 1995, p. 144-146

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