I am developing a program that needs me to determine which coordinate in a $2$-D figure has maximum distance from a fixed coordinate. Let me demonstrate:
$3$ points: $(1,3), (2,2), (5,0) $;
Fixed point: $(0,0)$ ;
Answer: $(5,0)$
I need to know which point on the Cartesian Plane has a greater distance from the origin WITHOUT calculating $\sqrt{a^2+b^2}$ (Euclidian distance) and preferably also without squaring and finding the square root.
Also, I need an approximation of this distance. Not necessary the literal distance (eg for $(3,4)$ I don't need $25$) but any transformed value in any domain which gives a measure of how further off the point is so I can use this value to compare to other maximum coordinates of other figures and quickly determine which coordinate will have maximum distance. A figure may have coordinates which have the same euclidean distance from the fixed point. (Eg $4,4$ and $-4,-4$).