Pole at the bank of a Circular Pond

Question

A pole stands at the bank of a circular pond. A man walking along the bank finds that the angle of elevation of the top of the pole from the two points A and B is 30 each and from the third point C, it is 45. If the distances from A to B and from B to C measured along the bank are 40 m and 20 m respectively; find the radius of the pond and the height of the pole.

In the book's solution, it is assumed that the mid-point of the AB arc, the center of the pond and the base of the pole are collinear. I wonder why it is so.

Answer 1

Let $P$ be the pole, and let $\overline{PQ}$, where $Q$ is opposite $P$, be a diameter. Note that the conditions of the problem imply that $AP = BP$; in such cases, $\overline{PQ}$ is a perpendicular bisector of $\overline{AB}$. If this doesn't seem clear to you, and the diagram below doesn't convince you, let me know and I'll amend with a sketch of a proof.

Warning: Diagram not to scale given the dimensions in the OP.

Answer 2

The rays from the top of the pole to the bank describe a slant cone with circular base. This is clearly symmetric with respect to the diameter through the pole basis. Therefore, if the angle is the same as seen from A and B, and since they are distinct, they must be symmetric to that diameter.