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Math Help - Prove base of tower is on line

  1. #1
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    Prove base of tower is on line

    \alpha, \beta, \gamma are the angles of the top of a vertical tower from three distinct points A, B, and C, on a horizontal line that is on the same level as the base of the tower but does not necessarily pass through it. The distances AB=BC=d. Prove that the height h of the tower is given by
    h^2(\cot^2\alpha-2\cot^2\beta+\cot^2\gamma)=2d^2
    If 2\cot\beta=\cot\alpha+\cot\gamma, prove that the line ABC passes through the base of the tower.
    I have already shown the h is given by h^2(\cot^2\alpha-2\cot^2\beta+\cot^2\gamma)=2d^2, but i don't how to prove that if 2\cot\beta=\cot\alpha+\cot\gamma the line passes through the base of the tower.
    Thanks for any help!
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  2. #2
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    If ABC line passes through the base of the tower O, and CO = x, then
    cotα = (2d + x)/h
    cotβ = (d+x)/h and
    cotγ = x/h.
    Now see what is (cotα + cotγ) =
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