So I have a problem I have been working on. I am trying to show that if I have $4$ points and $3$ of them are on a circle. Then the last point has to be on the circle if we say that the sum of distances from all the points are equal.
What I tried to use coordinate geometry but I do not see a nice way to do it. If we have the 3 points on a circle $a=(x_1,y_1),b=(x_2,y_2),c=(x_2,y_2)$ and the last point $d=(x_4,y_4)$ then we center the circle at $(0,0)$ and for simplicity set $r=1$ so we get
Now we can use this to simplify the distances from each point
Now we also know that $|ab|+|ac|+|ad|=|ab|+|bc|+|bd|=|ac|+|bc|+|bd|=|ad|+|cd|+|bd|$ from the fact that the sum of distances is equal.
Having all this information now I want to conclude that $x_4^2+y_4^2=1$
I know I can eliminate one of the unknown distances from the equalities but that does not seem to help.
Any input, hints would be appreciated.