Given a set of 3 2D points, how to find if they lie within a circle of a given radius? Given 3 points, A (x1, y1), B (x2, y2) and C (x3, y3), what is the best way to tell if all 3 points lie within a circle of a given radius r?
The best I could come up with was to find the Fermat point F (x4, y4) for the triangle ABC, and then check if the distance from F to each A, B and C is less than r. Is there a way to do this more efficiently?
 A: Welp, Rahul in a comment took the wind out of my sail.  Rather seriously.
Okay.  Measure the three distances.  Find the two that are furthest apart.  If that is more than $2r$ they are not in a circle.  If they are less than $2r$ apart, find the two points that are exactly $r$ away from both points.  Find two circles centered at the points with radius $r$.  If the third point is in either circle we are done.  All three points are within that circle.  If not, we are still done.  If the three points aren't within either of those circles they won't be within any other circles either.
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Take the perpendicular bisector of two pairs of points.  The intersection will be equal distance from all three.  (The three points will lie on a circle with that distance as a radius and that point of intersection as the center.)  If the distance if less than $r$ then the points are interior to such a circle.  Otherwise not. 
A: The smallest enclosing circle for three points is either the circumcircle of the triangle formed by the points (if the triangle is acute) or the circle whose diameter is the longest edge of the triangle (if the triangle is obtuse). The two circles coincide if the triangle is right-angled. There exists a circle of radius $r$ that encloses the points if and only if the radius of the smallest enclosing circle is less than or equal to $r$.
