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# Triangle inequalities help

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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

Jan 25, 2022

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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