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 T. Ono in 1914, the inequality is actually false; however, the statement is true for acute triangles and right triangles, as shown by Balitrand in 1916.

Statement of the inequality[edit]

Consider an acute or right triangle in the Euclidean plane with side lengths a, b and c and area A. Then

27 (b^2 + c^2 - a^2)^2 (c^2 + a^2 - b^2)^2 (a^2 + b^2 - c^2)^2 \leq (4 A)^6.

This inequality fails for general triangles (to which Ono's original conjecture applied), as shown by the counterexample a=2, \, \, b=3, \, \, c=4, \, \, A=3\sqrt{15}/4.

The inequality holds with equality in the case of an equilateral triangle, in which up to similarity we have sides 1,1,1 and area \sqrt{3}/4.

See also[edit]


  • Balitrand, F. (1916). "Problem 4417". Intermed. Math. 23: 86–87. 
  • Ono, T. (1914). "Problem 4417". Intermed. Math. 21: 146. 
  • Quijano, G. (1915). "Problem 4417". Intermed. Math. 22: 66. 

External links[edit]