# Ono's inequality

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

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 $\sqrt{3}/4.$

## References

• 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.