Proof for max. distance between two points inside a circle

Question

I have an algorithm in mind which can solve one of my algorithm's problems.

The algorithm links Blue-Dot to any of the Red-dot such that their mutual-distance is maximum compared to the distance between blue-dot and any other red-dot. (see the image below for a better understanding).

I am pretty much clear how to do that but I need more mathematical proves. I have done something on my part as well, but any help which makes it more evident mathematically will be highly appreciated. (I know the intuitive part well, but I need mathematical proof)

1. The central-node (black-dot in the centre, knows the cartesian coordinates of all dots and easily gets polar coordinates.
2. The circle is divided into four quadrants. This way Angle $\angle POR$ can easily be found.
3. It is for sure that the red-dot in the same quadrant is out-of-question since it will definitely have a smaller distance compared to other red-dots.
4. Using the law of cosines, length $r$ can be found for the rest of the three quadrants.
5. But, the ideal candidate is the exact opposite quadrant as following the law of cosine $r = \sqrt{a^2 + b^2 - 2ab\cos(\angle POR)}$ and in the opposite quadrant $\angle POR$ is very close to 180, hence $\cos(\angle POR) = -1$. This will make $r$ to be maximum. But I need mathematical proof or is this proof enough.