There is a wordy and confusing set-up.
At the end of the question the following appears,
The basic mathematical problem can be described as follows:
Given three points A , B , and C and the circle that goes through them, find the point P on the circle that maximizes PA+PB+PC .
In the comments, I made the following suggestion
The whole story of the runners just confuses things as evidenced by the answers so far. I suggest removing the preamble and simply stating the geometrical problem as at the bottom of the question. Note: We need to know whether distance is measured in straight lines or around the circle. – chasly from UK 6 hours ago
The answer came back
This a puzzle not a math problem site - so it is presented as a puzzle. As I stated some other place - it has to be straight lines. Along the circle P is coinciding with B. – Moti 5 hours ago
What do you think. To put it bluntly, who is right and who is wrong?