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Thread: One interesting formua

  1. #1
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    One interesting formua

    If $t_a,\ t_b,\ t_c$ are medians of triangle, $\alpha,\ \beta,\ \gamma$ interior angles, $P$ area of triangle, prove formulas:
    $${t_a}^2=\frac{1}{4}a^2+2P\cot\alpha$$
    $${t_b}^2=\frac{1}{4}b^2+2P\cot\beta$$
    $${t_c}^2=\frac{1}{4}c^2+2P\cot\gamma$$
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: One interesting formua

    We aren't going to do your homework for you, but we can help. How far have you got on these?

    -Dan
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  3. #3
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    Re: One interesting formua

    we know the following

    $\displaystyle 2\left(b^2+c^2\right)=\left(2t_a\right){}^2+a^2$

    $\displaystyle a^2=b^2+c^2-2 b c \cos \alpha$

    $\displaystyle P=\frac{1}{2}b c \sin \alpha$

    Eliminate $b,c$
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  4. #4
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    Re: One interesting formua

    Yes, this is solution. You forgot to mention formulas $abc=4PR$ and $\frac{a}{\sin\alpha}=2R$, where $P$ is area of triangle, and $R$ is radius of circumscribed circle.
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