Looking to find the distance from not the center of the earth to a point on the surface

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I am looking to find the distance from a point on the earth's surface to a fixed point within the earth, not at its center. This fixed point would not be on either of the earth's axis. It would be 10 km away. I want to code this into some software so it can update the distance as the earth rotates. I want to see how latitude affects the variation in distance. I have drawn a crude picture of what I am trying to achieve. I have done some searching on the internet to try and find something that works but I haven't really hit on anything yet.

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