Consider an acute triangle (meaning a triangle with all angles less than 90°) 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
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 all less than 90°, the tangent of each corner is positive, which means that the inequality above is correct by AM-GM inequality.
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- Lukarevski, M. (2017). "An alternate proof of Gerretsen's inequalities". Elem. Math. 72: 2–8.