I was going through "Challenging Mathematical Problems with Elementary Solutions (Vol 1)" by Yaglom & Yaglom.
I came across this problem,
Three points in the plane are given, not all on the same straight line. How many lines can be drawn which are equidistant from these points?
The solution states that if all three points are on the same side of the line, then they must lie on a parallel line, which is contradictory.
Thus, there are lines which exist such that points are on the same side. Thus, the book's answer is .
However, when I solved I got , a line perpendicular to the plane, through the centre of the circle, which passes through all the points.
Although the points are said to be in a plane, there was no constraint on the plane of the line.
Am I right? Or is it an accepted norm that lines are in the same plane as points?