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Ono's inequality

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In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by Tôda Ono (小野藤太) in 1914, the inequality is actually false; however, the statement is true for acute triangles, as shown by F. Balitrand in 1916.

Statement of the inequality

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Consider an acute triangle (meaning a triangle with three acute angles) in the Euclidean plane with side lengths a, b and c and area S. Then

This inequality fails for general triangles (to which Ono's original conjecture applied), as shown by the counterexample

The inequality holds with equality in the case of an equilateral triangle, in which up to similarity we have sides and area

Proof

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Dividing both sides of the inequality by , we obtain:

Using the formula for the area of triangle, and applying the cosines law to the left side, we get:

And then using the identity which is true for all triangles in euclidean plane, we transform the inequality above into:

Since the angles of the triangle are acute, the tangent of each corner is positive, which means that the inequality above is correct by AM-GM inequality.

See also

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References

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  • Balitrand, F. (1916). "Problem 4417". Interméd. Math. 23: 86–87. JFM 46.0859.06.
  • Ono, T. (1914). "Problem 4417". Interméd. Math. 21: 146.
  • Quijano, G. (1915). "Problem 4417". Interméd. Math. 22: 66.
  • Lukarevski, M. (2017). "An alternate proof of Gerretsen's inequalities". Elem. Math. 72: 2–8.
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