Distance between two points above horizon I know that the distance to the horizon is proportional to the square of the height of the person, $d=k\sqrt{h}$. If I'm given that a person is $1.8$ metres tall and sees $4.8$ kilometres out to sea, $k$ is $3577$. I can then pretty easily work out how far an $x$ meter tall person can see. However, if the $1.8$ metre tall person can just barely see the top of a ten metre mast above the horizon, how far away is the boat? How would I go about solving this problem? Thanks for your help.

if you can barely see it, then there's a spot on the Earth that is on the horizon for you _and_ for the boat: add the horizon distances for both yourself and the boat's mast.

![a diagram of the scenario: a circle with a tangent supported by two short lines](