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Haruki's Theorem

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Geometry Theorem
Illustration of Haruki's Theorem:
s 1 s 2 ⋅ s 3 s 4 ⋅ s 5 s 6 = 1 {\displaystyle {\frac {s_{1}}{s_{2}}}\cdot {\frac {s_{3}}{s_{4}}}\cdot {\frac {s_{5}}{s_{6}}}=1}

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 ⋅ s 3 ⋅ s 5 = s 2 ⋅ s 4 ⋅ s 6 {\displaystyle 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 {\displaystyle s_{1},s_{2},s_{3},s_{4},s_{5},s_{6}} are the measure of segments connecting the inner and outer intersection points.

The theorem is named after the Japanese mathematician Hiroshi Haruki.

References

  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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