# Proof that two objects on earth form a tangent line with the earth

So I have this problem where there is a building on earth at point $a$ and a robot with a camera on its head starts walking away from the building and I have to calculate what distance from $a$ does the robot stop seeing the top of the building. Now I have done my calculations and found the correct answer.

My problem lies in proving that the straight line distance between the building and the robot forms a tangent with the Earth. At the start of my work the line between the robot and building was assumed to create a tangent with our planet, which was correct as I did get the right answer. Now I have to prove that assumption but I dont know how.

Its obvious that it would create a tangent as the farther away the robot walks from the structure the closer its line of sight starts coming in contact with Earth's curvature. And at a certain point, at its maximum distance, it creates a tangent but by taking one more step back it loses sight with the building because the curvature of the planet would get in the way.

Hopefully that all made sense haha please let me know if it didn't.

I was thinking about saying something with upper and lower bounds but I feel like there's a much more simple way to explain it that I'm just not seeing. I'm not looking for a solution, just a clue or hint to point me in the right direction, if possible.

## 1 Answer

Right up until the point at which the building disappears from view, the line of sight has no intersections with the (assumed) spherical Earth. After that, the line of sight goes through the Earth, so has two points of intersection with the surface. At the point that the building vanishes, the line of sight intersects the surface at exactly one point. This by definition makes that sight line tangent to the surface.