I learned this method from the open math resources website. I couldn’t help myself repeat it in Rhinoceros. It was quite fun to solve circle tangency problems in 2D, this is one of them: drawing the circle that passes three given points, not using ready-made commands but only geometric tools of the circle (compass) and ruler (line). Here is the sequence of it:

First of all, we need to know that the circle we are looking for is centered somewhere on the perpendicular paths between the points. This means we should find the perpendicular directions of the triangle formed by the points. Two of these perpendiculars are enough because we’ll intersect them at the end. Also, these perpendiculars should pass the midpoint between pairs. First, draw a line between any two points:

You’ll have to find the perpendicular direction (+ the midpoint) between these two points. The easiest way to do it is by drawing equal radii circles and intersecting them. The radii of these circles might be anything greater from the mid-way between these points. If they are equal circles, this will give you the midpoint of these points.

Below you see how to construct the midpoint perpendicular of any two points in space. We also use this method to find the right angle direction while drawing the square:

Now, you have to do the same for another pair of points.

Just find the perpendicular line passing through the midpoint.

Just intersect these two perpendicular directions:

Here it is! The intersection of any two perpendicular lines will give you the center point of the circle that is equally far from every three points. It is like magic! :)

This also gives you the sphere that passes from three points (if you are not struck with Worldxy)