I'm questioning the solution of a high-school mathematics competition problem involving the geometric mean of distances from a point to a circle. The specific question is as follows:
Let O be the center of a circle with radius 4. Let A be a point outside the circle such that the distance from A to O is 14. Find the geometric mean of the distances from point A to every point on circle O.
For convenience, let O be centered at the origin, and let A be the point (-14, 0). The intended solution uses power of a point on point A to derive the result that the geometric mean of the lengths will be equal to the length of the tangent line from A to circle O.
https://i.stack.imgur.com/id1MJ.png (AD (blue line) is a tangent line from A to circle O)
The solution reasons as follows: For distances AB and AC, by power of a point, AB * AC = AD^2. The line can have greater or lesser slope arbitrarily. Thus, all the non-tangent points of the circle can be paired up in such a way that the geometric mean of the lengths from A to each point in the pair is AD. Clearly, the distance from A to a point of tangency is also AD. From common sense, the geometric means of all the relevant lengths will then be AD. By the Pythagorean Theorem, the length AD is sqrt(14^2 - 4^2) = sqrt(196 - 16) = sqrt(180) = 6 sqrt(5).
The reason I believe this is incorrect is because it seems to assign a heavier weight to points on circle O that are on the "A side". Specifically, the smaller arc length between the two point of tangency will be less than πr, while the larger arc length is greater than πr. But, each arc length contains an equal number of points (one from each pair) in the pairing solution above. This makes points more concentrated on the smaller arc (closer to A) for any finite number of lines drawn. I see no reason why this should not extend to an infinite number of lines including all the points of the circle. As an analogy, drawing many lines passing through the origin all with different slopes, for example with a angular separation of 1 degree between each line, will result in a much higher concentration of "line" near the origin than far away from the origin.
I have tried to think of a more rigorous solution for this, especially applying calculus-based techniques, but was unable to think of one.
Is the solution given really correct? If not, what is the correct solution?