# 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 F. 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

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

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

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