I'm posting this without having checked it in a reference, so I'm prepared to be full of s---.

This can be simplified to a problem in plane geometry, which I studied more years ago than I will ever admit. Here's the construct: Draw a circle of radius r. At the top of the circle, draw a tangent line. Draw a radius line between the center of the circle and the tangent point. Label this line "r". Draw another line from the center, out at an acute angle (30 degrees in my drawing) to intercept the tangent line. Label this line "h". Label the line "d" that goes from the tangent point to the intercept between h and the tangent line. You now have a right triangle rdh defined. The Pythagorean theorem tells us that h^2 = d^2 + r^2. Break the line h into two segements, r, between the center and the circle, and a, between the circle and tangent line. h = r + a. The altitude above the circle, where we observe the horizon is a. The distance to the horizon is d.

Substitute r + a for h, and solve the Pythagorean formula, and find:

d = square root (a^2 + 2ra).

The answer for r = 3900 miles and a = 35,000 feet is d = 228 miles.