We can find an acute triangle with the three altitude lengths \(1\)\(2\), and \(h\), if and only if \(h^2\) belongs to interval \((p,q)\). Find \((p,q)\).


Any help is appreciated thanks smiley

 Jan 25, 2022

I got it! The answer is \(({4\over5},{4\over3})\).


Here is how I got it:


Let S be the twice the area of the triangle. Then the side lengths are S, S/2, and S/h. Since the triangle is acute, these side lengths must satisfy these inequalities:

\(S^2 + ({S\over2})^2>({S\over h})^2\)

\(({S\over2})^2+({S\over h})^2 >S^2\)

\(S^2 + ({S\over h})^2>({S\over2})^2\)


We can divide each inequality by \(S^2\) to simplify, and the third inequality will automatically be satisfied, so we are left with:

\(1 + {1\over4} > {1\over h^2}\)

\({1\over4} + {1\over h^2} > 1\)


Simplifying, we get h^2>4/5, and h^2<4/3, so the interval for h^2 is the (4/5, 4/3).

 Feb 10, 2022

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