Calculating a Distance on the Surface of the Earth

I've managed to summarize the problem I'm trying to solve in the PDF
below. Basically, I have a point on the earth's surface at a given
latitude and longitude. If I moved 200km away from this point to the
East, I'd actually be above the surface of the earth due to the
curvature of the earth. If at the point, I were to drop straight down
to the surface of the earth, I'd like to know the distance between
this new point on the earth and the original point. I'd like to
repeat this calculation with heading 200km to the North. Help?

http://www.dafunks.com/misc/EllipseProblem.pdf
.

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