Relationship between the length of the tangent line through a point on sphere and great-circle distance As an aviator I'm familiar with the concept of great-circle navigation because when we fly a route between 2 points on the globe we know the shortest distance between these two points is the great circle distance.
I'm developing a navigation app in Google Earth and I need to calculate the shortest distance from the surface of the "spherical" Earth to any point on the tangent line through A (origin) when flying the great circle path.
Also, I'm using a mean earth radius of 6,371.009 km for WGS84 ellipsoid.
Just to be clear, I'd like to refer to the diagram in the following link:
http://www.alaricstephen.com/main-featured/2017/5/22/the-haversine-formula
I use the Haversine formula to calculate the distance, d, between the points A and D (see diagram). What I'd like to calculate is the distance D to E as a function of d.
In the diagram this is referred to as the external secant (exsec) which is the portion DE of the secant exterior to the circle.

HaverSine Formula is used routinely to compute long distances in navigation along shortest path great circles of the Earth between two points of given latitude and longitude.

First find $d$ on the Earth. Next air distance along a tangent if the flight point $B$ is above the Earth: $ t= r \tan \dfrac{d}{r}.$
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After clarification the above can be ignored.
For perfect sphere model of earth it is simple trig. calculation.
Distance $ AD= r \theta =  $ the arc distance you calculated using Haversine formula along a great circle arc of earth radius $=r$ as shown. Calculate $ \theta $ in radians in the plane of kite shape $OAEB$ if we imagine $B$ on another tangent point below. We have $  \theta= \dfrac{\text{arc} AD}{r}$

Central dimension is length $OE$. From this subtract earth radius.
$$ DE = r \sec \theta - r\; = r (\sec \theta -1 )$$
This is the red height above target/destination/landing place which should vanish on landing at $D$. It is indicated by exsec in the supplied link   for unit earth radius.
