I don't think that such a tool exists in ArcGIS. You could use other tools to precisely build your circle, but this would take you more time than just touching the three points. It took me one minute, so if I had to do plenty of circle I would certainly rather use a script. anyway, here is the workaround (based on the fact that there is only one circle passing through three points, and that the center of this circle is the intersection of the perpendiculars to the middle point of each segment joining the 3 points.
1) customize your editor toolbar by adding the "distance-distance tool" (drag and drop the distance-distance tool to your editor toolbar)
2) make sure that snapping is activated for your points
3) selection "polygon" in the construction tools
4) select "distance-distance" in the editor toolbar
5) make a polygon construction using the "distance-distance" tool
6) get the 2 intersections of the circles whose radius is the segment between two of your points (see illustration)
7) repeat that trick for another segment (doesn't matter which one)
8) select circle construction tool
9) place the center of the circle at the intersection of the two perpendiculars, and extend it until it snaps one of you points (it should then snap the three points. However, beware that pointing on the intersection of the construction will not use snapping. Therefore, if you need to be exact, it is better to terminate the polygon that you are drawing, then draw a the circle, and finally remove the polygon used for construction)
The figure below illustrates the3 points (triangle), the temporary polygon (in red) and the final circle (in grey). I have highlighted the two perpendiculars in yellow.