# Thread: Prove base of tower is on line

1. ## 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!

2. 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) =